UC-NRLF ; n n RYST/ LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Class THE OPTICAL INDICATRIX AND THK TRANSMISSION OF LIGHT IN CRYSTALS. LOXDOX : riHXTKD BY WILLIAMS AND STRAllAN, 7 LAWRENCE LAXE, CHEAPSIDK. THE OPTICAL INDICATRIX AND THE TRANSMISSION OF LIGHT IN CRYSTALS. BY L. FLETCHER, M.A., F.R.S., KEEPER OF MINERALS IN THE BRITISH MUSEUM ; FORMERLY FELLOW OF UNIVERSITY COLLEGE, MILLARD LECTURER AT TRINITY COLLEGE, AND DEMONSTRATOR AT THE CLARENDON LABORATORY, OXFORD. Jtonbon : HENRY FKOWDE, OXFORD UNIVERSITY PRESS WAREHOUSE, AMEN CORNER, 1892. CONTENTS. PAGE. INTRODUCTION . . . . . . ix. CHAPTER I. RECENT CHANGE OF VIEW AS TO THE PROPERTIES TO BE ASSIGNED TO AN ELASTIC LUMINIFEROUS ETHER. Deduction of the form of the wave-surface for biaxal crystals . . 2 Its singularities ........ 4 Dynamical difficulties of Fresnel's theory of double refraction . . 4 Results of the rigorous calculation of the vibratory motion of an elastic solid ........... 5 Lord Kelvin's version of the elastic theory .... 6 Fresnel's line of reasoning, and the terms based upon it, must be abandoned .''. . . . . . .7 The form of the wave-surface for biaxal crystals was really discovered in another way ....... 8 CHAPTER II. EVOLUTION OF THE OPTICAL INDICATRIX. General nature of light . . * -'-.. . . 9, Light is due to change of state of matter . . . .9 An ether is necessary . . . . . 9 Permeation of ordinary matter by the ether .... 9 The change of state is periodic . . . . . .10 The characters of an undulation . , . . . .10 Light is an undulatory phenomenon ...... . , . . 10 Sound is also an undulatory phenomenon . . . . .11 Intensity of light depends on the amplitude, colour on the period, of the vibration . . . . . . . .11 Polarisation of light : plane of polarisation : transverse plane . .11 Transmission of plane-polarised rays in glass and analogous media . 12 Geometrical representation of the characters of a ray of plane-polarised light ' . . .12 The laws of ordinary reflection and refraction accounted for by an un- dulatory theory . . . . . . .13 The laws of refraction of light by a crystal of calcite accounted for by an undulatory theory . . . . . . .14 The wave-surface is identical with the ray-surface J . . .14 Ray-front ......... 15 In the case of calcite, the ray-surface has two sheets, and consists of a sphere and spheroid . . . . . . .15 VI. CONTENDS. PAGE. Plane-polarisation of each of the refracted rays . . . .11) The plane of polarisation is related to the radius vector of the ray- surface . . . . . . . . .17 The above might have led to the recognition of the possible existence and optical characters of biaxal crystals . . . .17 Another mode of geometrically representing the characters of: thu extraordinarily refracted ray, by reference to the same spheroid, naturally presents itself . . . . . .18 The same mode also suffices to represent the characters of the ordinarily refracted ray without necessitating the use of a second surface . 19 The characters of the refracted rays can be simply expressed by reference to the spheroid alone . . . . .19 Generalisation . . . . . . . "20 The Optical Indicatrix ....... 20 .Relation of the optical indicatrix to the general symmetry of the crystal 21 CHAPTER III. NATURALNESS OF THE METHOD. Objections . . . ... . . .22 The development of Fresnel's theory '.,- . .. .23 Preliminary attempts at generalisation . ' : .,. .*, . . . 21 The history of the ray-surface . . . . . '. . 20 The true nature of the luminiferous ether . . . .' ^ .28 The educational difficulty . . . ' . ' . , - . 29 Three other modes of generalisation . . . . .29 Advantages of the method here suggested '.' . . . 30 CHAPTER IV. DEDUCTION OF THE OPTICAL CHARACTERS CORRESPONDING TO AN ELLIPSOIDAL INDICATRIX. ART. General Relation . . . . . . .31 1-3. The ray-surface : its construction, symmetry, and sections by the symmetral planes . . . . ." . .32 4-0. To find the characters of a ray in terms of the co-ordinates of-Zi 1 , the corresponding point on the indicatrix . . .30 7. The equation of the ray-surface . . . . .37 8-9. Given the direction-cosines of a line of transmission, to find (1) the velocities of the corresponding rays, (2) the co-ordinates of the corresponding points on the indicatrix . . .38 10. The corresponding points /ft, E^ are in a plane conjugate to the line of transmission . . . , . 9 11. To find their positions in the conjugate plane . . .39 12. Perpendicularity of the planes of polarisation of the two rays transmissible along the same line . . . .40 13. Op, O/ft, OR?., form a conjugate triad . . . .41 14. Given i\ and r. 2 , to find X p v. . . . . .42 15. Given the line of transmission, to find ri and r-j . . .43 10-'20. The optic bi-radials (secondary optic axes) . . .43 21-22. The ray-front corresponding to the ray Or is perpendicular to the transverse plane RNOr, and intersects that plane in a line parallel to OR . . . . . ... 48 23. The velocity of the ray Or, resolved normally to the ray-front, is measured, by the in verse, of OR. ... . ,52 CONTENTS. V.I. ART. PAOE. 24-5. OR is in general an axis of the section of the indicatrix by a plane parallel to the corresponding ray-front . . . 53 26. The two rays corresponding to a given direction of front- normal 54 27. The two front-normals corresponding to a given direction of ray . . . . . . .55 28-31. Inclination of a ray to its front . . . . .55 32. To find the direction-cosines of a front-normal in terms of the co-ordinates of R, the corresponding point on the indicatrix . 57 33. Given Imn, the direction-cosines of the front-normal, to find /! and/ 2 . . %. 58 34. Given/! and/ 2 , to find I m n . . . . .58 35-6. Given / m n or / and/ 2 , to find the co-ordinates of the corres- ponding points on the indicatrix . . . .59 Given I m w, to find the direction-cosines of the corresponding rays ........ GO Given X /* v, to find the direction-cosines of the corresponding front-normals . . . . . . .GO 39. The front-normal surface . . . . . .GO 40. The surface of wave- slowness or index-surface : it belongs to the same family as the ray-surface . . . . Gl 41-44. The optic bi-normals (primary optic axes) -. ; ,-f- . G2 45-52. The bi-radial and bi-normal cones . . . . G6 53. Representative surfaces derived from the indicatrix t . .71 CHAPTER V. VARIOUS OPTICAL RELATIONS WHICH ARE INDEPENDENT OF THE PHYSICAL CHARACTER OF THE PERIODIC CHANGE. 1. Object of the Chapter . . . . . . 73 2-8. Preliminary representation of the periodic change at any point of a ray of light ....... 73 9. Discrepancy of observed and calculated results . . .83 10-12. Its explanation ....... 83 13-16. A representative force : it is dependent on the luminous source . 85 17. A fallacy . 88 18. In general, if a plane-polarised ray is transmissible in a given direction, the plane of polarisation can have at most two different directions . . . . . .89 19. The refraction cannot be higher than double . . .90 20. Degree of the equation of the ray-surface . . . 90 21. The transmissibility of even a single plane-polarised ray is not a physical necessity : but if one position of a plane of polarisation is possible, there is a second at right angles with the first ........ 90 22. The transmission of a ray along an axis of tetragonal or hexa- gonal symmetry . . . . . . .91 23. The velocity- factor . . . . . .91 24. It is necessarily the same for all directions perpendicular to an axis of tetragonal or hexagonal symmetry . . .91 25. Transmission of a ray in a direction, lying in a plane of general symmetry, but oblique to an axis of tetragonal or hexagonal symmetry . . . . . . .92 2G. Transmission of rays along the axes of symmetry of an ortho- rhombic crystal , , . . , . ,93 Till. CONTENTS. ART. PAGE. 27. Transmission of rays in a syrametral plane of an ortho-rhombic crystal 93 28. Intersections of the ray-surface with the symmetral planes of an ortho-rhombic crystal . . ." . .94 29. General equation of the ray-surface for an ortho-rhombic crystal 94 30. The ray-surface for a mono-symmetric or anorthic crystal " . 95 31. The form of the ray-surface is independent of the physical character of the periodic change . . . .96 32. Resilience ........ ( J7 33-35. Free and forced vibrations : simple cases . . .97 36. Transmission of a simple forced vibration in an isotropically resilient medium . . . . . .98 37-8. More general cases of free or forced vibration of an seolotropi- cally resilient medium . . . . . .99 39. Transmission of a simple forced vibration in an seolotropically resilient medium ...... 101 40. Case of an ortho-rhombic crystal . . . . .102 41. Comparison with Fresnel's elastic forces .... 107 42. Case of a mono-symmetric or anorthic crystal . . . 107 43. An unsatisfactory variation of Fresnel's method . . .107 44. Transmission of elliptically or circularly polarised rays . 109 SUMMARY 110 INTRODUCTION. A SHORT account of the origin and development of this Tract, reprinted from the Mineralogical Magazine, may be of interest to the student. It is known that many years ago Mr. Maskelyne undertook the writing of a, Treatise on Crystallography. The book, embodying the mode of treatment of the subject adopted in his professorial lectures at Oxford and in a course of lectures given by him in the year 1874 to the Fellows of the Chemical Society, has been for some time complete as regards the purely crystallo- graphic portion, and the methods and nomenclature employed in it are now familiar to the students of Crystallography at the University of Cambridge and at the City and Guilds of London Technical Institute, South Kensington ; moreover, certain chapters on Crystallographic Physics have long been far advanced. But Mr. Maskelyne hesitated to publish his work without intro- ducing, in the case of certain of the problems of Crystallographic Physics, some more simple and satisfactory treatment than any hitherto suggested ; and, finding the subject too large to be satisfactorily dealt with in occasional hours of increasingly engrossing public occupations, he has from time to time invoked the aid of certain of his old pupils, among whom I have the happi- ness to be numbered. Some years ago I responded to a call of this kind, which involved the investigation of the behaviour of a crystal, viewed merely as an seolotropic body, in its relations to change of temperature. At that time the permanent rectangularity of a definite triad of lines of a mono-symmetric or anorthic crystal was very generally accepted by crystallographers. In two papers, read before the Crystallological Society (1879-83), this view was criticised, and the whole subject of the dilatation of crystals on change of temperature X. INTRODUCTION. was discussed by the aid of mathematics and general reasoning of an extremely simple character, as compared with any which had been recorded by previous workers. 1 The present Tract deals with another and more important problem, that, namely, of the behaviour of a crystal in respect to the refraction of light. The beautiful process invented by Fresnel has long been recognised, and increasingly so as time has gone on, as being dynamically unsound ; but there was no more rigorous method which did not involve mathematics of too high an order to be introduced into a book on Crystallography that should be generally useful, and none at all which was completely concordant with experiment in its results. The task proposed to me by my old friend and teacher was that of presenting the subject of refraction in a far simpler form to the student ; and, as an almost necessary condition for such a presentment, I had to endeavour to approach the subject by such a path as would render it possible to avoid, as much as possible, any discussion of the characteristics of the ethereal medium ; for physicists were still uncertain, not merely as regards the properties to be assigned to an elastic luminiferous ether, but even as regards the physical character of the pulsation which constitutes light. Meantime the problem assumed a different aspect ; for at the end of 1888 Lord Kelvin remarked that incompressibility, hitherto regarded as abso- lutely indispensable, is really unnecessary to the stability of the ether ; and he showed that the laws which determine the intensities of ordinarily reflected or refracted light are deducible from the properties of an ether which is compressible for the forces concerned in the transmission of light ; Mr. Glazebrook immediately followed with proofs that other important phenomena (such as ordinary and anomalous dispersion, double refraction, and metallic reflection) are likewise consistent with the new version of the elastic theory. Fresnel's process thus became more untenable than ever, even for the mere correlation of optical facts, for its hypotheses are completely at variance with those required by the new version. An elastic luminiferous ether is now to be regarded as compressible instead of incompressible ; its effective elasticity (of figure) is to be regarded as constant instead of variable, its effective density as variable instead of constant, for different media, and for different directions in the same medium if the latter be bi-refractive. But the mathematical development of the new version, involving as it does the idea of a slipless rigid boundary and a variable effective ethereal density, and the use of partial differential equations and triple integrals, calls for an amount of special knowledge which it is impossible to require from a purely crystallological student. It thus remained to invent, if possible, a process which should involve only elementary mathematics, be consistent with the 1 Lond. Dub. and Edinb. Philos. Magazine; 1880, ser. 5, vol. 9, p. 80: 18S3, ser. 5, vol. 16, pp. 275, 314, 412. INTRODUCTION. xi. hypotheses and results of the new version of the elastic theory, and yet be sufficient to serve the purpose of correlation of the phenomena of refraction, for which the method of Fresnel has been found of so great service. In the search for such an elementary process my attention was attracted to several remarkable facts : 1st. The direction now assigned to the ethereal vibration of a given ray is identical with that of Fresnel's elastic force. "2nd. The velocity of transmission of the ray is proportional to the magni- tude of that force. 3rd. The force is represented in direction by the normal of the "ellipsoid of elasticity," and in magnitude by the inverse of the length of the normal intercepted by the ray. It seemed that so simple a set of relations must be capable of translation by the aid of elementary mathematics into mechanical ideas consistent with the new hypotheses. An attempt to effect this took the form given on page 107, but was dis- carded as unsatisfactory. 1 became gradually convinced that all such attempts are premature, and that the adoption of any method of the kind would involve a possible recurrence of the present difficulty : for it is far from established that the new version, though incomparably more concordant with experimental results than the old, is anything more than a mere mechanical analogy, liable at any moment to be found incomplete, and to be replaced by a version of a totally different character. In fact, recent experiments seem to have established that light-waves and electro-magnetic waves only differ in length, yet the latter are deemed inexplicable as mere vibrations of an elastic ether. I3y this time, however, it had become manifest that a simple method of generalisation would have directly led to the discovery of the optical characters of biaxal crystals, without any reference to the specific characters of the ether at all ; but the difficulty remained that the general form of the wave-surface hud apparently been arrived at, not in this simple way, but by a priori reasoning founded on the properties of an elastic incompressible ether. This belief is a very general one, and is an almost inevitable result of a study of Fresnel's memoir, as published in the Transactions of the French Academy. That the belief is a mistaken one will be evident from the detailed history given in Chapter III. It must be remembered, in reading the memoir, that Fresnel, who died before its actual issue, was contending for the undulatory as against the emissive theory of light, and that most of his remarks are applic- able to vibrations in general as well as to the motions of the parts of an elastic ether. When it is made clear that Fresnel's deductive process was really an a posteriori one, and had not led to the discovery of the general form of the wave-surface, it is possible to part from his theory of double refraction with less reluctance ; and in adopting the method here suggested we shall merely be reverting to one which is at least analogous that by which his discovery Xll. INTRODUCTION. was actually made. And this can be done notwithstanding the admiration for Fresnel's brilliant researches which must be felt by every reader of his various memoirs. When those researches began, in 1815, the emissive theory of light was in the zenith of scientific favour ; when Fresnel died, in 1827, active opposition to the undulatory theory had virtually ceased : a result which was a direct consequence of Fresnel's reasoning and discoveries. As regards the method here suggested, it will be found that the idea of a correspondence between the characters of a ray and the geometrical characters at a point on an ellipsoid brings great simplicity into the study of the optical characters of crystals ; a simpler relationship than that which is taken as the basis of Chapter IV could not be desired. As for the mathematical development of that Chapter, it need only be pointed out that the equation of the ray-surface is deduced without the aid of the differential calculus or any complicated method of elimination : indeed, the Chapter requires no higher mathematical knowledge than is implied in the idea of conjugate diameters of an ellipsoid; the knowledge of infinitesimals demanded for the geometrical solution given in Art. 21 of that Chapter being of a very elementary character. The investigation of the geometrical rela- tions of Fresnel's wave-surface was long ago exhausted by Hamilton, Mac Cullagh, Sylvester, Pliicker, and Cauchy ; yet it is hoped that the adoption of the basis here suggested, the use of rays instead of waves, and the comparatively elementary character of the mathematical treatment given in Chapter IV, may enable the crystallological student to acquire a clearer idea of the geometrical relations of the wave-surface, and of the optical characters of crystals in general, than he is able to attain to by means of a method which is based on the hypothesis of varying ethereal density. It may assist the memory of the student if it is remarked that many of the symbolic letters used in the notation of Chapter IV are the initials of the corresponding words : r RI J? 2 p are all related to rays, N to a normal, f to a front, pi ^2 [p\\ [pi] TI ^2 and si s 2 [*i] [*a] ffi <*2 to the so-called primary and secondary optic axes respectively. In the last and more difficult Chapter the object in view is a different one : it is there sought to deduce the general form of the wave-surface by elemen- tary reasoning from simple hypotheses relative to the general characters of undulations, and without assuming that the vibration is an actual motion due to ethereal elasticity. L. FLETCHEL'. Louden, J/a/vv 7 / olst, 1891. ON THEORIES OF LIGHT. " Most thinkers of any degree of sobriety allow that an hypothesis of this kind is not to be received as probably true, because it accounts for all the known phe- nomena. But it seems to be thought that an hypothesis of the sort in question is entitled to a more favourable reception, if, besides accounting for all the facts pre- viously known, it has led to the anticipation and prediction of others which expe- rience afterwards verified. Such predictions and their fulfilment are, indeed, well calculated to strike the ignorant vulgar. But it is strange that any considerable stress should be laid upon such a coincidence by scientific thinkers. If the laws of the propagation of light accord with those of the vibrations of an elastic fluid in as many respects as is necessary to make the hypothesis a plausible explanation of all or most of the phenomena known at the time, it is nothing strange that they should accord with each other in one respect more. Though twenty such coinci- dences should orcur, it would not follow that the phenomena of light are results of the laws of elastic fluids, but at most that they are governed by laws in some mea- sure analogous to these ; which, we may observe, is already certain, from the fact that the hypothesis in question could be for a moment tenable. Who knowsbut that some third hypothesis, including all these phenomena, may in time leave the un- dulatory theory as far behind as that has left the theory of Newton aud his suc- cessors ? " (Mr. J. S. Mill, 1843-5 1 .) " We are led to the conception of a complicated mechanism capable of a vast variety of motion, but at the same time so connected that the motion of one part depends, according to definite relations, on the motion of other parts, these motions being communicated by forces arising from the relative displacement of the con- nected parts in virtue of their elasticity. The agreement of the results seems to show that light and magnetism are affections of the same substance, and fhat light is an electro-magnetic disturbance propagated through the field according to electro- magnetic laws." (Prof. J. Clerk Maxwell, 1865.) " I only mean that if light, as is generally supposed, consists of transversal vibra- tions similar to Ihose which take place in an elastic solid, the .vibration must be normal to the plane of polarisation. '1 here is unquestionably a formal analogy between the two sets of phenomena extending over a very wide range ; but it is another thing to assert that the vibrations are really and truly to-and-fro motions of a medium having mechanical properties (with reference to small vibratiocs) like those of ordinary matter." (Lord Rayleigh, 1871.) " While the elastic-solid theory, taken strictly, fails to represent all the facts of experiment, we have learned an immense amount by its development, and have been taught where to look for modifications and improvements. Nor is it surpris- ing that a simple-elastic-solid theory should fail. The properties we have been considering depend on the presence of matter, and we have to deal with two sys- tems of mutually interpenetrating particles. It is clearly a very rough approxima- tion to suppose that the effect of the matter is merely to alter the rigidity or density of the ether. The motion of the ether will be disturbed by the presence of the matter ; motion may even be set up in the matter-particles. The forces to which this gives rise may, so far as they affect the ether, enter its equations in such a way as to be equivalent to a change in its density or rigidity, but they may, and probably will, in some cases do more than this." (Mr. E. T. Glizebrook, 1885.) " It follows that the luminiferous ether is experimentally shown to be the medium to which electric and magnetic actions are due, and that the electro-mag- netic waves are really only very long light-waves. If magnetic forces are analogous to the rotation of the elements of a wave, then an ordinary solid cannot be analo- gous to the ethdr, because the latter may have a constant magnetic force existing in it for any length of time, while an elastic solid cannot have a continuous lota- tion of its elements in one direction existing within it. The mctt eatiefactory model, with properties quite analogous to those of the ether, is one consisting of wheels geared with elastic bands." (Prof. G, F. Fitzgerald, 1890.) CHAPTER I. RECENT CHANGE OF VIEW AS TO THE PROPERTIES TO BE ASSIGNED TO AN ELASTIC LUMINIFEROUS ETHER. Deduction of the form of the wave-surface for biaxal crystals. FRESNEL'S representation of the laws of transmission of rays of light in biaxal crystals, by reference to the surface distinguished by his name, has long been regarded as one of the greatest achievements in the domain of Physical Science. In his memoir 1 on Double Refraction, Fresnel proceeded as follows : 1. He assumed that the transmission of a ray of light is effected by means of an elastic ether vibrating transversely to the ray-direction. To the ether is thus assigned a property not belonging to a perfectly fluid body in a state of rest : perfect fluidity of a body at rest involves incapacity of resistance to mere change of shape, and it is to such dis- tortional resistance that transverse vibrations must be due. 2. He assumed that the ether of a crystal, when undisturbed, is a system of equal particles, in stable equilibrium under their mutual attractions ; and that, for each pair of particles, the attraction depends solely on some func- tion of the distance between them and acts in the line joining the centres. He showed that in a medium so constituted there are at least three directions, at right angles to each other, such that the force necessary to the maintenance of a small displacement of a single particle of the ether along any one of them will act in the line of the displacement, and be proportional to it in magnitude : that the elastic force evoked by the displacement of a single particle of the ether through unit- distance along each of these directions may be different, say a 2 , & 2 , c 2 , respectively : that in this case, which is assumed to be that of the i-Memoires de VAcad. de I'lnstitut de France, 1827, vol. 7, pp. 45-170. DEDUCTION OF THE WAVE-SURFACE FOR BIAXAL CRYSTALS. 8 ether in a biaxal crystal, the elastic force due to the displacement of a single particle of ether in any direction distinct from the three already mentioned will act in a direction different from that of the displacement ; that if the direction of a radius vector of the surface a?x 2 + 6 a // 2 -|- cV = (* l2 + .V 2 + 2 ) 2 represent that of the displacement of an ethereal particle, and the corresponding elastic force for a displacement through unit- dis- tance be resolved along and perpendicular to the line" of displacement, the former component is proportional to the square of the radius vector in magnitude : that for displacements of a single particle in directions lying in a given plane passing through the centre of the above surface, the elastic force is generally obliquely inclined to the plane, but that there are always two directions, namely, those of the longest and shortest diameters of the section of the above surface by the given plane, for which the resolved component of the elastic force in the given plane acts in the line of displacement. 3. He assumed that the ether is virtually incompressible for the forces concerned in the transmission of light. Neglecting, therefore, the component of the elastic force normal to a plane containing a set of similarly displaced particles (wave-front) as being without effect by reason of the incompressibility of the ether, Fresnel in- ferred that, for particles in the given plane, vibrations parallel to either the longest or shortest diameter of the corresponding section of the above surface must be persistent, since the only effective component of the elastic force for each particle then acts in the direction of the displacement. 4. From a suggested but forced analogy of a line of vibrating ether- particles to a vibrating string, Fresnel assumed that the velocity of trans- ference of a wave-front along its normal is directly proportional to the length of that principal diameter of the section of the above surface by the wave-front which is parallel to the direction of vibration. Hence finding, by the usual mathematical process, the envelope of planes representing the positions to which wave-fronts, with every possible direc- tion, would arrive after the lapse of the same interval of time, Fresnel concluded that the wave-surface for a biaxal crystal is represented by the equation 0: further, as the front corresponding to any ray is parallel to the tangent plane to the wave-surface at the point where the ray meets it, the vibra- tion is, in general, obliquely, not perpendicularly, transverse to the direc- tion of the ray. 4 THE TRANSMISSION OF LIGHT IN CRYSTALS. 5. Hence Fresnel also inferred that the velocities of the two rays which can be transmitted along a given direction are directly proportional to the 3* ?/ 2 ,.2 axes of the ellipse in which the ellipsoid ---f y 2 + = 1 is intersected by (I u C" a plane normal to the common direction of the rays. Singularities of tlie form. The closed surface represented by the above equation is of very peculiar form, and consists of two concentric ellipsoid-like sheets, which are symme- trical with respect to three rectangular planes. There are four points com- mon to both sheets ; they are situated at the extremities of two diameters lying in one of the planes of symmetry : in the neighbourhood of each of these points the sheets are drawn towards each other, and the surface has there the shape of a double cone ; an infinite number of tangent planes to the surface can thus be drawn at each of them. Further, two planes and their parallels respectively touch the surface, not at one point nor at two points, but at an infinite number of points which lie on the circumference of a circle. These geometrical singularities of the wave-surface, first noticed by Fir William Hamilton five years after the death of Fresnel, point to the existence in biaxal crystals of certain optical characters which had up to that time remained undiscovered, and seemed too strange to be real : the establishment of their actuality by Lloyd has been regarded as the crowning triumph of Fresnel's theory of double refraction ; for not only are the phenomena strange, but their observation demands a combination of circumstances which places them beyond the range of accidental dis- covery. Dynamical difficulties of FrestieVs theory of dnulle refraction. As continued experiment and precise observation have served only to establish the high degree of accuracy of the form assigned to the wave- surface by Fresnel, 1 it might naturally be inferred that the assumptions which lead, after so elaborate a course of reasoning, to a surface presenting these singularities must be themselves beyond cavil. Yet, strange to say, the mathematical process, by which the surface is thus arrived at, is one of which the weakness was recognised by the author himself, and the 1 Kohlrausch: Wied. Ann.; 1879, vol. 6, p. 86 ; vol. 7, p. 427. Glazebrook: Phil. Trans.; 1879, vol. 170, part 1, p. 287. Proc. Roy. Soc.; 1883, vol. 4, p. 393. DYNAMICAL DIFFICULTIES OF FRESNEL'g THEORY. 5 theory has long been regarded as dynamically unsound ; further, the characters assumed for the ether, though they lead to the true wave- surface, have since been found to have for necessary consequences other optical laws which are inconsistent with the results of experiment. On the other hand, the same form of wave-surface can be arrived at from other Bets of assumptions, which have thus the same claim to recognition; 1 yet they are inconsistent with those of Fresnel, and with each other. As the lator hypotheses which lead to FresnePs wave-surface have been found to have other consequences which are contradicted by experimental results, the comparative simplicity and the historical interest of the method of Fresnel have sufficed to secure the adoption of his assumptions and corresponding terminology in the general literature relating to the optical characters of crystals. The fact that Fresnel's wave-surface has been deduced from several inconsistent sets of assumptions as to the characters of the ethereal motion suggests that the form may really depend on the feature common to all, namely, the transmission of a periodic change of state differently related to different sides of the ray, and be otherwise independent of the physical character of the transmitted change : the suggestion is discussed in Chapter V. Eesitlts of the rigorous calculation of the vibratory motion of an elastic solid. The rigorous calculation 2 of the vibratory motion of the parts of an iso- tropic elastic solid is found to involve two quantities, which are generally denoted by A and B : the latter, Z>, measures the rigidity, or the resistance of the body to simple change of shape, or the elasticity of figure ; the former, A, is connected with B, and with k (which measures the resistance to simple change of volume, or the elasticity of volume), by the relation k=A B. Further, it can be shown that a vibratory motion of the parts of an elastic medium generally gives rise to two kinds of waves, due respectively to distortional and condensational-rarefactional vibra- tions ; the former travelling with velocity \/ , the latter (which correspond to those of sound) with velocity N/ , where p is the density of the medium. Now, if the transmission of light through a singly refractive medium be 1 e.g. Challis in the Trans. Camb. Phil. Soc.; 1847, vol. 8, p. 524. 2 A most Valuable Report by Mr. Glazebrook on Optical Theories is published in the Hep. Brit. Assoc. for 1885, pp. 157-261. G THE TRANSMISSION OI-' LIGHT IN CKYtTALS. duo to the vibratory motion of an isotropic elastic solid, all the energy persists in the form of distortional vibrations perpendicular to the ray ; hence the characters of the ether must be so assumed as to secure the absence of the condensational-rarefactional vibrations. For this purpose we may make either of two assumptions, namely, that A is virtually zero or that A is virtually infinite as compared with B : in the former case the condensational-rarefactional wave is got rid of by making its velocity zero ; in the latter case by making the velocity infinite. But it was long believed that the former assumption was otherwise inadmissible : for it was supposed by Green and later mathematicians that the quantity A B is necessarily positive, if the equilibrium of the parts of an elastic body is stable ; and this is impossible if A is zero, for B is essentially a positive quantity : hence it only remained to assume A and therefore also k infinite, and thus the ether to be virtually incompressible. Double refraction could then be consistently explained by a variation of the rigidity of the ether of a bi-refractive crystal with the direction ; but ifr was necessary, for dynamical reasons, to assume the vibrations to be in, not perpendicular to, the plane of polarisation. On the other hand, Lord Rayleigh 1 has proved that the phenomena due to the scattering of light by small particles require the vibrations of the ether to be perpendicular to the plane of polarisation ; he has further shown that no theory based on varying rigidity can possibly be satisfactory, and that the variation of density in different directions in a biaxal crys- tal would lead dynamically to a form of wave-surface different from that of Fresnel, if the ether be incompressible for the forces involved in the propagation of the vibrations. Lot\l Kelvin's version of the elastic theory. From this position of dead-lock, according to which the ether must be both compressible and incompressible, the theory that the transmission of light is effected by the vibrations of an elastic medium has only recently been extricated. At the end of 1888 Lord Kelvin, 2 re-examining the problem of the stability of the equilibrium, found that the condition that A Ii is a positive quantity becomes unnecessary, " provided we either suppose the medium to extend all through boundless space, or rive it a fixed containing vessel as its boundary :" with either of these provi- sions, the stability only requires that A should not be negative, and it is therefore possible to get rid of the condensational-rarefactional wave by 1 Lond. Eclin. and Dub. Philos. Magaz., 1871, ser. 4, vol. 41, p. 451. a Hid., 1888, ser. 5, vol. 26, p. 414. ANOTHER VERSION OF THE ELASTIC THEORY. 7 the assumption hitherto deemed inadmissible, namely, that A is zero ; this involves the compressibility of the ether for the forces concerned in the propagation of light. As a mechanical illustration, Lord Kelvin points out that " homogeneous air-less foam held from collapse by adhesion to a containing vessel, which may be infinitely distant all round, exactly ful- fils the condition of zero-velocity for the condensational-rarefactional wave ; while it has a definite rigidity and elasticity of form, and a de- finite velocity of distortional wave, which can easily be calculated with a fair approximation to absolute accuracy." Starting with the new assumption, Lord Kelvin was able to deduce correct expressions for the intensities of ordinarily reflected or refracted light : and Mr. Glazebrook 1 has since shown that the elastic theory in its new form fully accounts for dispersion, including anomalous dis- persion (like that of cyanin), double refraction, and metallic reflection, and further that it leads to a correct expression for the velocity of light in a moving medium. According to the new version, the vibrations of the ether are perpendicular to the plane of polarisation, even in biaxal crystals, and thus always perpendicularly transverse to the ray : further, the matter- particles and ether-particles are supposed to react on each other : and if their vibrations are synchronous, the former may even be set in appreciable motion by the latter. As the reaction of the matter and ether may produce the same effect on the motion of the ether-particles as would result from a simple variation of the rigidity or density of the ethereal medium, it be- comes convenient to distinguish between the actual and effective values of the rigidity and density. It is clear that the new version of the properties of the elastic ether, whether really true or not, 2 is far more satisfactory than any hitherto sug- gested, and must replace the older versions until a better one is proposed. Hence it becomes necessary, for those who adopt an elastic ether as the basis of the undulatory theory, to regard (1) the ether as compressible, even for the forces concerned in the propagation of light ; (2) the actual density and rigidity of the ether as identical for all bodies ; (8) the effective rigidity as invariable ; (4) the effective density as different in different bodies, and, in the case of doubly refractive crystals, in different directions within the same body. FresneVs line of reasoning, and the terms based upon it, must be abandoned. For the great majority of mineralogical students, the chief value of the 1 Ibid., 1888, ser. 5, vol. 26, p. 521. 3 Ibid., p. 538 ; 1889, vol. 27, pp. 24G, 253 : Nature, 1889, vol. 40, p. 32. 8 THE TRANSMISSION OF LIGHT IN CRYSTALS. hypothesis of an elastic ether is in the correlation of the phenomena observed when light is transmitted through crystals ; for which purpose it is very desirable that the student should be able to reach the wave-surface, if practicable, by means of elementary reasoning based on observed facts of a simple character. The rigorous calculation of the motions of a vibrating elastic medium is not a simple process : it involves, indeed, mathematics of so high an order that the derivation of the wave-surface in this way will always be unintelligible to the ordinary student of crystals. On the other hand, the only comparatively simple mode of derivation of the wave-surface, as yet invented, that of Fresnel, depends upon assump- tions of incornpressibility and varying elasticity which are now deemed untrue ; and further, involves for biaxal crystals a general obliquity of transverse vibration, not in accordance with the latest version of the elastic theory. Under present circumstances, the process of Fresnel, even if adopted on account of its great historical interest, must be puzzling to the student, and inevitably lead to the acquisition of wrong views as to the properties to be assigned to the luminiferous ether ; hence it becomes necessary to abandon the whole process, and all those terms now in com- mon use (ellipsoid of optic elasticity, axes of optic elasticity, coefficients of optic elasticity) which are based upon it. The form of the wave-surface for liaxal crystals was really discovered in another way. The great difficulty in the correlation of the phenomena of the transmission of light through biaxal crystals, as already stated, lies in the derivation of the wave-surface. The form of the surface is too extraordinary to be directly assumed either as a probable one a priori, or as suggested by ex- perimental results. If it can be shown that the form of the wave -surface for biaxal crystals is suggested by a simple generalisation, independently of any particular version of the undulatory theory, and might have been brought in this way within the province of experimental investigation, the greater part of the present educational difficulty will be removed from the path of the student. In fact, we shall find that it was really by a process of generali- sation, though not indicated in the composite memoir of 1827, that Fresnel himself was first led to the true form of the wave- surface for biaxal crystals. The properties of an incompressible elastic ether were mathematically developed by him after the discovery of the true form of the wave-surface had been made. ^ CHAPTER n. EVOLUTION OF THE OPTICAL INDICATUIX. lu the present Chapter it is sought to show that a certain surface, here termed the Optical Indicatrix, naturally suggests itself as a means of correlation of the laws of transmission of light in uniaxal crystals ; a simple generalisation then suggests the possible existence of biaxal crystals, and the general nature of their optical properties. The reasoning may be arranged as follows : General nature of light. Light travels with finite velocity. A flash of light transmitted from one body to another may thus for a time be wholly in the intervening space ; hence the transmission of light must be one either of matter or of change of state of matter. Light is due to the change of state of matter. Two rays of light of the same colour, travelling in the same direction along the same line, may annihilate each other. Hence the transmission of light cannot be one of matter ; it must be a transmission of change of state of matter, and the change must be capable of representation by positive and negative quantities. An ether is necessary. Light travels across interplanetary space. Hence interplanetary space must be filled with one or more kinds of matter, capable of transmitting a particular kind of change of state with an enormous but finite velocity (18G,000 miles a second), and for distances amounting to millions of millions of miles. We may conveniently assume that the extraordinary matter is wholly of one kind, and designate it by a special name, ether ; it must be extremely subtle, for it offers no appreciable resistance to the motion of the planets. Permeation of ordinary matter by the ether, Light is transmitted, but with different velocities, through ordinary matter. 10 THE TRANSMISSION OF LIGHT IN CRYSTALS. Hence either ordinary matter is itself capable of transmitting this particular kind of change of state, or it is permeated by an ether capable of so doing. Having regard to the enormous velocity with which light is propagated through interplanetary ether and different kinds of ordinary matter, we may assume that the same kind of ether is concerned in the transmission, and that the variation of velocity and other characters is due to the influence of the ordinary matter on the properties of the permeating ether. The change of slate is periodic. If two rays, continually transmitted along the same line, annihilate each other, annihilation again takes place if either ray is transferred through any multiple of a certain measurable distance along the direction of trans- mission. Hence, so long as a single ray of light is being transmitted along a line, the state of the ether at a given instant is the same at all points distant from each other by a certain measurable quantity, which we may denote by X. But the continual uniform transmission of the change of state along the line involves a continual and periodic change of state at each point of the line ; the duration of the period being the same at all points, and always equal to the time necessary for the transmission of the change through the distance X along the line : if v be the distance of transmission during the unit of time, the period will thus be . During a single period, the ether at any point in the line of transmission experiences all those changes which belong at a given instant to all points in a length X of the line of transmission. The characters of an undulation. "Whatever be its physical nature, a periodic change of character at any point is termed a vibration of the character: its maximum value, the amplitude of the vibration : the interval of time required for a complete vibration, its period: the state at a given instant, the phase of the vibration : the relation between the phase and the time, the law of the vibration. If, further, the change is being transmitted along a line or ray, the con- figuration of the states at all points of the ray at a given instant is termed an undulation : the least part of an undulation which includes all varieties of phase is termed a wave, and the distance occupied by a wave, a wave- length . Light is an undulatory phenomenon. It follows from the above that, in this general sense, light is undoubtedly an undulatory phenomenon of some kind or other. POLARISATION : TRANSVERSE PLANE. 11 Sound is also an undiihttory phenomenon. By similar reasoning, it follows that sound is an undulatory phenomenon. Experiment shows that the transmission of sound is effected by ordinary matter, and that the change of character is one of oscillation of the material p.r.licles, the oscillation being generally solely in the direction of the transmission. The properties at any point of a line of transmission of a c:>:itiiiuocl uniform sound, namely intensity, note and timbre, must de- pond on the characters of the vibration at the point, and thus on the amplitude, period and law : experiment proves that the intensity of a simple sound depends solely on the amplitude, and the note solely on the period. Intensity of light depends on the amplitude, colour on the period, of the vibration, Similarly, the corresponding properties at any point of a ray of ordinary light, intensity and colour, may be assumed to depend on the characters of the vibration at the point, and thus on the amplitude, period and law: v,*c may tentatively assumo, from analogy with sound, that the intensity of a simpb ray depends solely on the amplitude, the colour solely on the period. Polarisation of litjht; plane of polarisation : transverse plane. But common light is capable of a change to which there is no parallel in the case of sound. A ray of common light transmitted through air acquires distinctive characters by reflection at a certain angle of incidence from a shoet of glass : as tested by reflection at the same angle of inci- dence from a second plate of glass, it has different properties on different sides ; its properties being symmetrical, however, at every point of the p?,t.k to the same two perpendicular planes intersecting in the ray : one of the planes of symmetry is the plane of incidence and reflection from tho first plate. As the planes of symmetry of the ray are dissimilar and can bo experimentally distinguished from each other, that which coincides with the plane of incidence and reflection may conveniently be termed the plane of polarisation; the second plane of symmetry may be dis- tinguished as the transverse plane. A ray having the same characters, however induced, is said to be plane-polarised. Hence the periodic change of the ether at any point of an aerially transmitted plane-polarised ray of light is not solely related to the direction of transmission, and thus differs in kind from that which charac- tjrises sound. For the suggestion of the laws of double refraction, prcciser knowledge of the character of the change is unnecessary. 12 THE TRANSMISSION OF LIGHT IN CRYSTALS. Transmission of plane-polarised rays in glass and analogous media. If a plate of ordinary glass or any analogous medium be placed with its faces perpendicular to an aerially transmitted plane-polarised ray, the light which emerges from the glass is found to be still plane-polarised, and the position of the plane of polarisation is found to be unaltered whatever the thickness of the plate : this is still true, if the plate be turned through any angle round its own normal. As the direction of the ray within the plate is coincident with the direc- tion of the ray before incidence and after emergence, we may thus reason- ably assume that, at all points of the line of transmission within the plate itself, the periodic change of the ether is symmetrical to the same two planes ; in which case the position of the symmetral planes of the periodic change is wholly independent of the glass and depends only on the direc- tion of the plane of polarisation of the incident ray. A plane-polarised ray transmissible in any direction within such a medium may have any azimuth of plane of polarisation whatever. Geometrical representation of the characters of a ray of plane-polarised liyht. In representing the transmission of a ray of plane-polarised light, of a I FIG. 1. single given colour and given intensity, within a given medium, we have thus three characters to consider : 1. The line of transmission of the ray, 2. The direction of the plane of polarisation, 3. The velocity of transmission. The direction of a plane being most conveniently denned by the direction of its normal, the above three characters may be geometrically repre- sented by means of two intersecting perpendicular lines, one of them definite in position, the other only in direction : and any definite function of the length of either may represent the velocity. The direction of transmission r } and the plane of polarisation p y r HUYGENS'S CONSTRUCTION. 13 of a given ray (Fig. 1), may thus be represented by two lines Or, T^Y; where R N is any line perpendicular fco the plane p q r, and therefore also to the ray r : the velocity of transmission may be represented by any function of either of the lines Or, RN~. If 0, a point on the ray, be given, and the normal of the plane of polarisation be taken to intersect the ray, all the characters may be represented by means of a single line R N, not passing throwjk the yicen point : for the line r is then known, since it passes through and intersects RN perpendicularly. The laws of ordinary reflection and refraction accounted for ly an undid atory theory. Two hundred years ago (1678-90), Huygens showed, by reasoning which is really independent of the physical nature of the periodic change, though he imagined it to be identical in character with that involved in the trans- mission of sound, that the laws of ordinary reflection and refraction of light are compatible with an undulatory theory. He assumed that a general disturbance of the ether at any given point must eventually produce disturbances at all other points of the medium, and that in a transparent body showing ordinary refraction the velocity of transmission of the dis- turbance is independent of the direction ; all points on a spherical surface having the given point for centre are thus at any moment in a similar state of disturbance. If we have regard to the arrival of the disturbance from its origin, we may say that in this case the front of the disturbance at any epoch is a sphere. The front of the disturbance due to a single centre may, for the sake of brevity and generality, be called the u-ave- surface. If the disturbance at the centre be persistent and periodic, the surface which defines the front of the disturbance at a given epoch passes through points of the medium at which, notwithstanding the continual change at each point, there is persistent identity of phase of vibration. Huygens gave a geometrical construction for the determination of the direction of the refracted ray by means of the spherical wave -surface, the direction being that of a line joining the point of incidence of the ray to the point of contact of a tangent plane of the wave-surface, drawn through an auxiliary line which lies in the refracting surface and is normal to the plane of incidence : if v be the velocity in the first medium, i the angle of incidence of the ray, and the size of the sphere correspond to the lapse of a unit of time, the distance of the auxiliary line from 1! the point of incidence is sin i 14 THE TRANSMISSION OF LIGHT IN CRYSTALS. The laws of refraction of light by a crystal of calcite accounted for by an undulatory theory. In the case of calcite, the refraction is in general not single but double. One of the rays, and only one, follows the laws of ordinary refraction for all directions : hence Huygens inferred that the surface of disturbance corresponding to this ray is the same as for ordinary media, namely, a sphere. It was necessary to assume a different form of surface of dis- turbance to account for the extraordinary refraction of the other ray, and the surface which first suggests itself, after a sphere, is an ellipsoid : further, since the refraction of the second ray is the same for all directions equally inclined to a special direction in the calcite-crystal, or since rays lying in a plane perpendicular to this line obey the laws of ordinary refraction, it is necessary for the surface of disturbance to be one of revolution about that direction as axis. Testing this hypothesis and finding it satisfactory, Huygens inferred from his observations that the surface of disturbance corresponding to the second ray is really a spheroid, touching at the extremities of its axis the spherical surface of disturbance corres- ponding to the first ray. This relation between the surfaces has been confirmed by later experimenters and found to hold for other crystals analogous to those of calcite : it is undoubtedly a Law of Nature. 1 The direction of the extraordinarily refracted ray is given by the same geometrical construction as before, the surface of disturbance being taken as a spheroid instead of a sphere. The wave-surface is identical with the ray-surface. Let rs, ES, (Fig. 2) be two wave- surfaces due to an origin 0, and with the line Or R as axis describe a cone of small angle, determining areas a b, AB, upon them. There is great difficulty in imagining the exact nature of the physical process by which an isolated ray could be propagated through the ether by means of undulations : still the conception of a ray of light comes so naturally, and has been found so serviceable from the very earliest times, that rays, rather than waves, will be used throughout the present Tract. Having regard to the apparent rectilinearity of propagation within a homogeneous medium, we may reasonably assume that, if light is propa- gated by the disturbances of a medium and the disturbances at all parts of 1 Stokes : Proc. Roy. Soc., 1872, vol. 20, p. 443 ; Comp. Rend., 1873, vol. 77, p. 1150. Abria: Ibid., p. 814. Glazebrook : Phil. Trans., 1880, vol. 171, part 2, p. 421. Hastings: Amer. J. Sc., 1888, ser. 3, vol. 35, p. 60. BAY-FRONT. 15 the first surface are allowed to produce their effects at the second surface, the resultant disturbance of the area A B is identical with that which would directly follow from a rectilinear transference of the disturbances at points on the area a b to corresponding points on the area A B ; and thus that the length Or, which represents the distance to which the front of disturb- ance has travelled in the direction Or in a given interval of time, also represents the velocity of transmission of a ray of light in the same direction. Regarded from this point of view, the surfacs of disturbance or wave-surface may be termed the ray -surf ace. Bay-front. Further, if a pencil of rays having OrR for axis starts simultaneously from 0, the front of the pencil at a certain epoch is a portion of the ray- surface containing r, and at a subsequent epoch is a portion of the ray- FIG. 2. surface containing R : in the limiting case, where the pencil is of extremely small angle, its front is in the tangent planes at r and Pi at successive epochs. The plane front, which thus belongs to an extremely small pencil including a given ray, may be briefly denoted as the ray -front for that Since the ray- surface retains a constant similarity of form and position, for the ratio OR : Of depends solely on the time, the tangent pianos at R and r are parallel. When the ray-surface is not a sphere, the tangent plane at any point is in general inclined obliquely to the radius vector drawn to the point from the origin, and a ray-front is then oblique to its corresponding ray. In the case of calcite, the ray-surf ace has two sheets and consists of a sphere and a spheroid. Huygens was thus led to the discovery that the laws of refraction in 16 THE TRANSMISSION OF LIGHT IN CRYSTALS. the case of calcite are consistent with an undulatory theory, if the velocities of transmission of rays of light within this mineral are determined by a sphere and a spheroid, touching each other in the axis of revolution of the latter. If a line Or z i\ (Fig. 3), drawn from the common centre 0, intersects the sphere and spheroid in r a and i\ respectively, according to Huygens the velocity of transmission of one ray in the direction O/yj is measured by 0/a, and of the other by 0/v For a single direction of Orj\, namely that of the axis of revolution COC, the two points r., and i\ coincide, and the rays travel with equal velocity ; this direction is called the optic axis of the crystal. Plane- polarisation of cadi of Hie tcf rented rays. Po far we have had regard merely to the relation of the two velocities to the direction of ray-transmission within the calcite-crystal. It did not escape the notice of Huygens, however, that each of the rays emergent from a crystal of calcite differs from common light : the difference is one which he was unable to account for by his version of the undulatory theory. It was not till more than a century afterwards (1808) that Malus made the accident -il discovery that the same change in the character of the light may be induced by reflection from a plate of glass : to this altera- tion, termed by Malus polarisation, the attention of physicists was largely directed during the immediately succeeding yeais. If a plate of calcite be placed with its facts perpendicular to an aerially transmitted plane-polarised ray, the latter is in general divided at the first surface into two : the two rays travel through the plate in directions mutually inclined to each other, and, emerging from it, are transmitted through the air with the same direction as that of the original ray : each of the emer- gent rays is plane-polarised, but the planes of polarisation of the rays have not the same direction ; they are, in fact, perpendicular to each other. Further, when the plate is turned round its normal through any angle, the plane of polarisation of each emergent ray is also displaced through exactly the same angle : the direction of the plane of polarisation is thus dependent on characters belonging to the plate itself: it is found to be in- dependent of the direction of the plane of polarisation of the ray incident on the plate. For one of the emergent rays, the line of transmission within the plate is continuous with the path of the ray before incidence and after emer- gence : as the position of the plane of polarisation of the emergent ray is independent of the thickness of the plate, we may reasonably assume, as PLANE OF POLARISATION AND THE RAY-SURFACE. 17 before in the case of glass, that the ray transmitted ivithin Hie plate is like- wise plane-polarised, and that the plane of polarisation during such trans- mission is identical in direction with that of the emergent ray, thus rotating with the plate as the plate is turned round its normal. As the emergent rays are indistinguishable from each other in charac- ter and only differ in the positions in space of their planes of polarisation, we may likewise assume that the second emergent ray is also transmitted within the plate as a plane-polarised ray, but with a direction of plane of polarisation perpendicular to that of the first. It will be found that the characters of rays which have been transmitted through a plate of calcite can be accounted for, if we imagine that in such a medium a plane-polarised ray t ansmissiblc in any given direction has its plane of polarisation in one or other of two rectangular positions, which depend on the crystal itself. . As in the case of air, glass, and analogous media, the periodic change of the ether at every point of a plane-polarised ray transmitted within any bi-refractive medium may be assumed to be dissimilarly symmetrical to two perpendicular planes ; but it may be remarked that it is only the disturbed ether which is assumed to be dissimilarly symmetrical in the distribution of its characters. 1 The plane of polarisation is related to the radius vector of the ray-surface. Malus 2 discovered that the direction of the plane of polarisation of any ray transmitted within a crystal of calcite is determined by the direction of the corresponding radius vector of the ray-surface : he showed that the plane of polarisation for the ray Oi\ (Fig. 3) corresponding to the spheroid is always perpendicular to the plane Or^C, which contains the ray-direction and the optic axis, and for the ray Or 2 corresponding to the sphere is the plane Or.,C, which also contains the ray-direction and the optic axis : in other words, the plane containing the ray-direction and the optic axis is the plane of polarisation of the ray belonging to the sphere and the transverse plane of the ray belonging to the spheroid. The alove might have led to the recognition of the possible existence and the optical characters of biaxal crystals. The above facts and reasoning were known to physicists before the exis- tence of biaxal crystals had been discovered ; further, the reasoning is really independent of the physical nature of the vibratory change which constitutes light. We proceed to prove that though the geometrical 1 See also pages 7, 79. 2 Mem, pres. a Vlnstitut ; Paris, 1811, vol. 2, p. 413 18 THE TRANSMISSION OF LIGHT IN CRYSTALS. representation of the laws of transmission of light in biaxal crystals was suggested to Fresnel by ideas in which elasticity had a great part, the possible existence of such crystals, and the corresponding laws of transmission of light, might have been deduced from the above by a simple generalisation, involving no reference either to the constitution of the luminiferous ether or to the nature of the physical change involved in the transmission of light ; and further, that the step was so natural a one to take that the discovery of the true form of the wave -surface for biaxal crystals could scarcely have been long avoided. Another mode of geometrically representing the characters of the extra- ordinarily refracted ray, by reference to the same spheroid, naturally presents itself. Draw ORi parallel to r^V (Fig. 3), the tangent at t\ to the ellipse in which the spheroid is cut by the plane r^OC\ OE l and Oi\ are said to be conjugate to each other, and the tangent at Pi t is parallel to Qr t . By a well-known property of the ellipse, the area of the parallelogram of which OE lt Or,, are adjacent sides is constant, whatever the direction of 0r,: hence the area is OA'OC; OA and OC being the principal axes of the ellipse, and therefore conjugate to each other. But if 7?^ is perpendicular to O,, meeting it in N l9 and is thus normal to the ellipse and therefore also to the spheroid at R lt the area of the parallelogram is also Or^P^N^ Hence Or, = OA'OC whatever the direction of Or^, audOA'OC bein" 1 a constant quantity, the velocity of a ray transmitted in the direction Oi\ may be represented, not only by Oi\, but by the inverse of B l N l . But the REPRESENTATION BY WEANS OF THE SPHEROID ALONE. 10 same lino -ZijiYj determines the plane of polarisation of the ray Or^ for as already stated, the line li^i is normal to that plane. Further, the same line R^i determines the direction of the ray, for the ray passes through and is perpendicular to TijAV Hence the direction, velocity and plane of polarisation of the ray Or, can all be represented by means of a single corresponding line J^A^, which is at once normal to the spheroid and the ray. This moth of representation naturally presents itself as soon as the plane of polarisation is indicated by its normal ; in fact, any attempt to represent geometrically the observed facts of the double refraction of cal- cite almost inevitably leads to it. The same mode also suffices to represent the characters of the ordinarily refracted ray without necessitating the use of a second surf. ice, But for any radius vector Ot\ of a spheroid there are always two normals of the spheroid which intersect it perpendicularly : one of them has just been indicated, namely HiN^\ the other is normal to the plane ^OC, at the centre of the spheroid, and therefore always lies in the equatorial plane. As already stated, the plane of polaruation of the ray Or. 2 is r z OC or r^OC : and the normal of its plane of polarisation thus lies in the equatorial plane, and is normal both to the spheroid and the ray. Further, the intercept made by the ray Or a upon this normal of the spheroid is OA, whatever the direction of Or 2 r 1 : hence, if the same law as before holds for the relation of the velocity to the intercept upon the normal of the spheroid, the velocity of the ray Or z is TTJ~> or ^C; and this is exactly the velocity required. Hence the velocity and plane of polarisation of the ray Or 2 can likewise be represented by means of a corresponding line which is at once normal to the spheroid and the ray : and this line indicates the plane in which the ray having these characters will lie. The characters of the refracted rays can be simply expressed by reference to the spheroid alone. All the characters of rays transmitted in various directions through a crystal of calcite may thus be simply expressed by means of a tingle sur- face, the spheroid. The relation of the optical characters of the crystal to the geometrical characters of the spheroid is as follows : To every given point on a single surface, a spheroid, there in general corresponds one ray : the direct ion of the ray is that of a diameter 20 THE TRANSMISSION OF LIGHT IN CltYSTALS. intersecting perpendicularly the normal drawn at the point to the spheroid ; the velocity of the ray is inversely proportional to the length of the normal intercepted between the surface and the ray ; the plane of polarisation of the ray is perpendicular to the same normal. For points of the spheroid lying on the equatorial circle or at the ends of the axis of revolution, the normal passes through the centre, and the direction of the ray becomes indeterminate : if such a point be re- garded as the limiting case of a small ring, it corresponds, not to a single ray, but to an infinity of rays lying in a plane perpendicular to the normal, all transmitted with the same velocity, and all having the same plane of polarisation. In the case of singly refractive substances there is a spherical surface of reference for which the same general relations are true. Generalisation. But it immediately suggests itself that in the case of a crystal like barytes, of which the morphological development and the physical charac- ters are dissimilarly symmetrical to three rectangular planes, the surface of reference, if such a surface exists, is more likely to be an ellipsoid with three unequal axes than an ellipsoid of which two axes are equal. In fact, the correspondence of the optical and the morphological symmetry of crystals was announced by Brewster in 1819. In the fourth Chapter are deduced the laws of transmission of light in a crystal for which the surface of reference is an ellipsoid having three unequal axes ; starting with the hypothesis that the relations between the geometrical characters of the surface of reference and the optical characters of the medium are identical with those which have just been found to obtain when the surface of reference is either a spheroid or a sphere. The Optical Indicatrix. To the surface of reference the term Optical Indicatrix may be assigned : this suggestive term has the advantage of being equally applicable whether the surface of reference is an ellipsoid, a spheroid, or a sphere, and it is independent of all versions of the undulatory theory ; the adjectival prefix may be omitted when the term Indicatrix involves no ambiguity. The Indicatrix is identical in form with the ellipsoid of elasticity of various authors, the ellipsoid of polarisation of Cauchy, the ellipsoid of indices of Mac Cullagh, and the index -ellipsoid of Liebisch, RELATION TO THE GENERAL SYMMETRY. 21 Relation of the optical indicatrix to the general symmetry of the crystal. In regard to the arrangement of its faces, every crystal is found to belong to one or other of six types of symmetry, distinguished as cubic, tetragonal, hexagonal, ortho-rhombic, mono-symmetric, and anorthic : further, it has been demonstrated by the mathematician that the types of crystalline symmetry thus met with are precisely those which are pre- sented by systems of planes of which the relative positions can be ex- pressed by means of whole numbers, a law to which the faces of crystals are found to conform. Further, we are led by experiment to the induction that a type of symmetry is such, not only for the arrangement of the faces of a crystal, but for all the physical characters : the planes of symmetry characteristic of the types are thus planes of general symmetry. On the other hand, a plane may be one of symmetry for a particular character without being a plane of general symmetry of the crystal : the type is thus not necessarily determinable from the symmetry of the crystal with respect to a single character. For example, a crystal may have the six faces of a cube and really belong, not to the cubic, but to the tetra- gonal, or even the ortho-rhombic type ; observation of some character other than the geometrical being thus necessary to the distinction : again, a plane inclined at any angles to the planes of general symmetry of a cubic crystal, and any plane containing the morphological axis of a tetra- gonal or hexagonal crystal, is a plane of symmetry for the changes pro- duced by dilatation on change of temperature, and is generally not a plane of symmetry for the facial arrangement. The above induction requires a plane of general symmetry to be a plane of symmetry of every indicatrix : on the other hand, a plane of symmetry of a single indicatrix is not necessarily a plane of general symmetry of the crystal. Hence, if the most general form of the indicatrix be an ellipsoid, it will follow that in the case of an ortho-rhombic crystal the axes of any indica- trix must coincide with the three axes of general symmetry. For a tetra- gonal or hexagonal crystal, the symmetry of the indicatrix with respect to the general planes of symmetry requires two of the axes of the ellipsoid to be equal, and the ellipsoid to be one of revolution about the morpholo- gical axis. For a cubic crystal, the symmetry of the indicatrix with respect to the general planes of symmetry necessitates the equality of all the axes of the ellipsoid, and the surface becomes a sphere. The above is true for all colours of the light, though the relative mag- nitudes of the axes, both for the general ellipsoid and the ellipsoid of revo- 22 THE TRANSMISSION OF LIGHT IN CRYSTALS. lution, may vary with the colour : further, it is true for all temperatures of the crystal consistent with the stability of the structure, for a plane of general symmetry must retain that character between the assumed limits of temperature. In the case of a mono-symmetric crystal, the induction still requires the plane of general symmetry to be a plane of symmetry of the indicatrix for all colours of light and for all temperatures consistent with crystalline stability ; but the positions and dimensions of the two axes of th,e ellipsoid lying in the plane of general symmetry are otherwise independent of the latter, and will in general vary both with the colour of the light and the temperature of the crystal. And in the case of an anorthic crystal, in which there is a centre, but no plane, of general symmetry, the positions and dimensions of all three rec- tangular axes of the indicatrix corresponding to a given colour or tempera- ture are free from limitations by a plane of general symmetry, and will l : kewise vary both with the colour of the light and the temperature at which the determinations are made. CHAPTER ill. NATURALNESS OF THE METHOD. Objections. To the above reasoning, by which it is sought to prove that in the case of calcitc the reference of the two sheets to the spheroid alone is one which it is natural to make, and not a mere geometrical artifice only to be discovered after the truth of the generalisation has been established, it may be objected that the reference would in such case have been made long before the present century. It must be remembered, however, that the consequent generalisation would have been a barren speculation at a timo when the polarisation of light by reflection was still undiscovered (1808), and the optical characters of most doubly refractive crystals were still beyond the powers of observation ; indeed, it was not till the decade 1310-20 that any scries of numerical data were available for the testing of DEVELOPMENT OF FRESNEI/S THEORY. 23 a theory : even the accuracy of the construction given by Huygens for the determination of the directions of the refracted rays in calcite was discredited by most physicists at the beginning of this century. But it may be fairly objected that if the above reference and generalisa- tion were natural, the discovery of the process would have preceded the development of any elastic theory of double refraction. And, in fact, it was really by a process of generalisation that Fresnel's discovery of the true form of the ray-surface for biaxal crystals was made. When the above argument was written, the detailed history of Fresnel's theory had not come to the notice of the author : as the facts are not generally known, and have an important bearing on the true significance of the clastic theory of double refraction, it becomes desirable to explain the position. The development of Fresnel's theory. Fresnel's celebrated memoir on Double Refraction was not printed till 1827: in that year, and before the issue of the memoir, Fresnel died at tho early age of 89, after years of illness. In the memoir are incorporated papers submitted to the Academy at different dates in the years 1821 and 1822, and it occupies no less than 132 pages of large size. For the sake of brevity, Fresnel made many omissions from the papers as origin- ally submitted to the Academy, and for the sake of clearness adopted a synthetic mode of treatment : the result is that the memoir as printed gives no clue to the real order of discovery, and the reader is apt to infer that Fresnel discovered the true form of the ray-surface a priori by means of equations relative to the elastic forces evoked by the disturbance of an incompressible elastic ether. The following statement by Aldis 1 exem- plifies this, which is still a very general impression : " Fresnel's theory is undoubtedly not a sound dynamical theory. It has, however, the great merit of representing accurately the facts of double refraction as far as experiment at present has tested them, and in one instance has led to the discovery of facts (the conical refractions) pre- viously unobserved. Probably, when the Newton of Physical Optics has succeeded in linking together all the phenomena of Light into one continuous chain, the name of Fresnel will yet be remembered with a reverence akin to that which astronomers feel for Copernicus and Kepler." The real order of development was of course known to some of Fresnel's 1 \V. S. Aldis. A Chapter on FnsneVs Theory of Double Refraction* Cambridge, 1870, p. 26. 24 THE TRANSMISSION OF LIGHT IN CRYSTALS. contemporaries, but to the next generation it was a mystery ; it was not till forty years after Fresnel's death that the mystery was dispelled by the publication of the original memoirs, which had been carefully preserved in the family. Yerdet, 1 one of the editors of Fresnel's collected papers, makes the following remarks : " It may seem odd that reasoning which .is incomplete and inexact in two points should have for result one of the best confirmed of the Laws of Nature. But we have seen that this law became manifest to Fresnel as the result of a generalisation quite similar to the generalisations which have led to most great discoveries. When he wished afterwards to account for the law by a mechanical theory, it is not astonishing that he should have led the theory, perhaps unwittingly, towards the end which he already knew of, and that, in his choice of hypotheses, he should have been de- termined, less by their intrinsic probability, than by their agreement with what he was justified in believing to be true. We have seen some traces of the progress of his ideas in the marginal notes which he had added to the manuscript of memoir No. 38, a memoir here printed for the first time. In the later memoirs we find nothing but the explanation, in different forms, of the mechanical theory by which he tried to demon- strate a posteriori the laws which direct intuition had revealed to him." After this clear statement on the part of his editor, it is obvious that Fresnel's theory of double refraction, however ingenious, has no claim t3 credit for its predictions : ths latter are really a direct consequence of the generalisation which had preceded the theoretical development of the vibratory properties of an elastic but incompressible ether. Preliminary attempts at generalisation. The first attempt at the generalisation of Huygens's construction had sug- gested a sphere combined with a concentric ellipsoid having three unequal axes as the most general form of ray-surface : this assumed that in the most general case one of the rays obeys the ordinary laws of refraction. It was found, however, that the refraction of the second ray as experi- rnent.illy determined is inconsistent with an ellipsoidal form of ray- surface. Nor would such a combination of ray-surfaces account for the optical characters of a biaxal crystal : for if a concentric sphere and ellipsoid meet each other, they must either touch at the extremities of a principal diameter, or intersect in two curves ; in the former case there 'would be only one direction of equal ray- velocity; in the latter case this (Euvrcx Competes iVA. Frcsncl : Paris. 18*58, vol. 1', p. 327. TWO EMPIRICAL LAWS DISCOVERED BY EIOT. 25 character would belong to every diameter which passes through the curyes of intersection, and thus to an infinity of lines lying on the surface of a cone. In 1819 Biot male the important discovery that the results of optical measurement are consistent with two empirical laws, both of them reached by processes of generalisation : in combination with the assump- tion that one of the rays obeys the laws of ordinary refraction, they completely express the polarisation and velocity of the second ray in terms of its direction in the crystal. 1 1st lau\ We have seen that in the case of a uniaxal crystal, two rays transmitted along any given direction had been shown by Malus to have their planes of polarisation respectively coincident and at right angles with the plane containing the ray-direction and the optic axis : from this Biot was led by generalisation to the discovery that in the case of a biaxal crystal the planes of polarisation are the internal and external bisectors of the angle between the two planes which contain the ray-direction and pass each through one of the optic axes. 2nd laic. ID the case of a uniaxal crystal, if i\ and r a be the velocities of transmission of the two rays transmissible in a direction inclined at an angle a- to the optic axis, it follows from Huygens's construction that the ratio ( ., -, h sinAr is constant for all directions : noticing this, 'Biot \'-'i v- J was led by generalisation to the discovery that in the case of a biaxal crystal the ratio ( a " ):sin o^ sin o-. 2 is constant; o^, cr. 2 , being the v'r ^'2 / inclinations of ihe ray to the optic axes. The second law, combined with the assumption that the velocity of one of the rays is independent of its direction, leads to a surface of the fourth degree, tangent to a concentric sphere at the ends of two diameters, as the i ay-surface corresponding to the second ray. For, let I n, 1 ??, \ ju v,be direction-cosines of the optic axes and of the second ray respectively (Fig. 9) : then cos o- x = / X + n v, and cos cr a = / A + n v. If r be the variable velocity of the second ray, and a be the constant velocity of the first ray, it follows from the above law that : 2 = k sin Oj sin o-^, where k is a constant. 1 Memuins de VAca.il. de VInstitut de France, 1820, vol. 3, pp. 228, 233. 26 THE TRANSMISSION OF LIGHT IN CRYSTALS. 1/1 1 \ 2 Hence pl^-^l =(l+coser,)(l cos s.>. y. For any direction of ray lying in a plane oi symmetry, there are thus two possible directions for the plane of polarisation, perpendicular to each other ; and in general each plane of polarisation corresponds to a different velocity of transmission in the given direction. For two directions of the ray, those of the diameters s^,, S 2 S 2> the two velocities are equal (Fig. 7c). If v, z be the co-ordinates of one of the points s, we have X 2 2 ~2 + a = 1 and or 5 -f = b 2 : hence c * (a? - b 2 ) a? (6 2 - c 2 )* If X v be the direction-cosines of a diameter Os, the relation may also be written as The angle sflC is given by the relation III I}' v v> 2 ~v/ h. In each of the planes OBC, OCA, OAB, a plane -polarised ray is thus transmissible with the velocity a, 6, or c, respectively, whatever its direction in the plane: hence, by the general principle of undulations, the refraction of these rays by a surface perpendicular to the symmetral v v v plane will be ordinary, and the index of refraction will be -, -, -, respec- tively; v being the constant velocity of transmission in the other medium. If a, /3, y, be the values of the index of refraction of the rays in each of the above planes, for which the index of refraction is independent of the direction, we must have 1 1 1 a:b:c=-: -:-; P y a,/3, y, are termed the ptineipal indices of refraction. 36 THE TRANSMISSION OF LIGHT IN CRYSTALS. 4. Given the co-ordinates x'y'z' of E, to find the velocity r of the corresponding ray Or. The velocity of the ray Or is measured by -. (Fig. 5) : but EN, being the normal of the indicatrix at the point E and perpendicular to OX, by construction is equal to the perpendicular drawn from the origin to the plane aVa? + tfy'y + c^z'z = 1 which touches the indicatrix at E. If p be the length of this perpendicular - = V(aV 2 + by + cV 2 ). P Hence r 2 = ^ =^ 2 = V 2 + &y* + cV 2 : r being the length of that radius vector of the ray- surf ace which corresponds to the point E of the indicatrix. 5. Given the co-ordinates x' y' z' of E, to find the direction-cosines of the normal of the plane of polarisation of the corresponding ray Or. The plane of polarisation, being perpendicular to the normal EN (Fig. 5), is parallel to the tangent plane of the indicatrix at the point E ; the equation of the tangent plane may be written in the form PO?X'X -\-pb z y'y -}-pc <2 z'z=p. Hence the direction-cosines of the normal of the plane of polarisation are , pb*y' t r ' r ' where r 2 6. Given the co-ordinates x'y'z' of /?, to find the direction-cosines of the corresponding ray Or. The direction- cosines of ^Y (Fig. 5) being - , , 7, and x' y' z 1 those of OE being -, ^, -, where r' = OE, the direction-cosines hkl of a line perpendicular to both EN and OE, and therefore to the plane EON and all lines therein, are given by the equations: hence y'z'(b*-c 2 )~z'x' (c 2 -a 2 EQUATION OF THE RAY-SUEFACE. 87 If Xjj, v be the direction-cosines of any line whatsoever in the plane liON, or ys Let the line X //, v coincide with Or, in which case it is at right angles to aV by cV Ju\ r , of which the direction-cosines are , , ; we thus have Determining the ratios X : /u : v from the last two equations, we get X v where A = ajW^-o 8 ) - a B = y'd*a*(a*-P) - y (7 = *y 2 & 2 (& 2 -c 2 ) - * X r in which r a = aV a +6y a +cV a , as before. These equations determine the direction- cosines Xju v of the ray Or cor- responding to the point x'y'z' or B. 7. The equation of the ray-surface. The co-ordinates of r being x y z, we have#=\r, y=/ur, 55= vr : Xpv being the direction-cosines of the ray Or. Hence, substituting these values in the last set of equations, _ Each of these fractions is equal to But the denominator of the last expression is zero ; for by construction the line Or, of which the direction-cosines are -, , , is perpendicular to EN, of which the direction-cosines are , , (Art. 5). Hence the numerator is also zero ; for the fractions equivalent to the expression are never all of them indeterminate, and are never infinite. 38 THE TRANSMISSION OF LIGHT IN CRYSTALS. this is the equation of the ray-surface, for it expresses a relation between the co-ordinates xyz of any point r lying in it. The equation may also be written in the form or 8. Given \^v, th? direction- cosines of a line of transmission, to find r, and r 2 , the velocities of the corresponding rays Or 1} Or. 2 . Substituting the values x=\r, y = \i r i z vr in the equation of the ray- surface, we have 6V - *V and, multiplying out, r 4 (a 2 A 2 +&V 2 + c2 ^-' 2 {a 2 (& 2 +c 2 ) \ 2 + fc 2 (c 2 +a 2 ) /t a + c a (a 2 This being a quadratic equation in r 2 , there are in general, for given values of X /* v, two solutions, say r x 2 and r 2 2 , and thus two velocities. of transmission in the given direction. A geometrical solution is given in Art. 15. The above equation may sometimes be conveniently written in the form 9. Given X p v, the direction-cosines of a line of transmission, to find the co-ordinates x y r/, ar a ' y z of the points R lt R 2 , of the indicatric ichith correspond to the rays Or^ Or 2 , respectively. Having found r? and r 2 2 , as indicated in the last Article, the co- ordinates Xi yi Ki t A' a ' ?/./ z.J are given by the equations (Art. 6) : ^ v 1 i sa y - and \ u r r.? a* r a a 6 2 r./ c 2 remembering that and tf since the points 7?j, 7? a , are on the indicatrix. THE POINTS CORRESPONDING TO A GIVEN RAY-DIRECTION. 89 10. The points E^ 71 2 , corresponding to the rays Ot\, O 2 , transmissible along theline \pv, are in a plane conjugate to that line. Since the normals of the indicatrix at R 1} R^ are both perpendicular to the line X p, v we have Art V +/u&V + vcV =0. Hence the points R^ R%, are in the plane This is the equation of a plane passing through the centre of the indicatrix and parallel to the planes which touch the indicatrix at either of the points where the line X p v intersects it. It is also obvious geometrically that the tangent planes at R^ R% t are both of them parallel to Or, and that Or is therefore parallel to their line of intersection ; Or is thus conjugate to the plane containing the points 0, fl lf R 2 . And it is geometrically evident that at all points of the section made by the conjugate plane the tangent planes to the indicatrix are parallel to, and therefore their normals perpendicular to, the conjugate line Xjuv: the points R lt RZ, are those of the section for which the normals of the indica- trix are not only perpendicular to the line \pv, but intersect it. 11. Given the direction of transmission, tojind the positions of the cor* responding points R lt R^, in the conjugate plane. From the last Article, it follows that the tangent planes to the indica- trix at its intersection with the conjugate plane form a tangent cylinder, having its axis parallel to the direction of transmission. Let UKVbe the curve of contact of the cylinder and indicatrix (Fig. 8) : R^ RZ, are some- where on the curve UKV. As a line is only perpendicular to its conjugate plane when it coincides with an axis of the indicatrix, Or, the axis of the cylinder, is in general oblique to UKV, the conjugate plane. Let U'K'V', U"K"V" be sections of the cylinder by two planes per- pendicular to its axis ; they are in general ellipses : let K"KK' be any line on the cylinder parallel to the axis, and K"L", KL, K'L', be the normals of the cylinder at the points K", K, K', respectively ; they are evidently parallel to each other. But KL is also the normal of the indicatrix at K, for the cylinder and indicatrix are tangent to each other at that point : also K'L' lies in the plane U'K'V , since that plane is perpendicular to the axis of the cylinder : hence K'L' is the normal of the ellipse U'K'V' at the point K'. 40 THE TRANSMISSION OF LIGHT IN CRYSTALS. The line KL will thus only intersect the axis of the cylinder when K'L' is an axis of the section U'K'V. Hence the points 2? lf 7? 2 , Ji^, /i } 2 , arc the four positions of A", on the curvo UKV, for which the normal of the inclicatrix intersects the axis of the cylinder : and these four positions are projections, by lines parallel to the axis of the cylinder, of the extremities of the axes of its " base ; " the base being taken as perpendicular to the axis of the cylinder. In other words, the points E lt E^ and the normals -fi^Yj, E 2 X 2 , lie in the planes of symmetry of that tangent cylinder of the indicatrix which has its axis in the common direction of transmission of the rays. u" u V Flo. 8. 12. The planes of polarisation of the two rays Or lt Or% t transmissible along the same line are perpendicular to each other. By the last Article, the normals 7? x A 7 ",, Z? 2 A T 2 , are in the planes of sym- metry of the tangent cylinder and at right angles to its axis : hence the planes of polarisation, to which the two lines are perpendicular, are them- selves perpendicular to each other. The following analytical proof is interesting by reason of the elimina- tions : The normals of the planes of polarisation being normals of the indicatrix at li lt li. 2) their direction-cosines are a 2 .r. 2 ' i 2 //./ c 2 2., f -, T -,, - / , respectively (Art, 5) ; '2 '2 '2 hence, if be the angle between the planes of polarisation, A TRIAD OF CONJUGATE DIAMETERS. 41 Substituting tlio values of ar/y^s/i # 2 'y 2 '~2 f > from Article 9, we have >S Now r/ 2 , ?./, being the roots of the equation given in Article 8, we have ft2 ^ and fiV. 1 - - hence (^- l( ' 2 )(r^-^) = rfa*-a* (rf+rf) + * Similarly, and _, W - 2 ) (^ - 2 ) (r^ - ft 2 ) (r 2 2 - ft 2 ) * (r x 2 - ^ 2 ) (r 2 2 - C ' 2 ) and is a right angle. 13. // p be the point in icliich Or intersects the indicatrir, the lines O/o, ORi, OR 2 , form a triad of conjugate diameters of the indicatrid'. The axes of the basal section U'K'V (Fig. 8) of the tangent cylinder of the indicatrix being conjugate to each other, it follows, from the pro- perties of parallel projection, that the projections of the axes on any section of the cylinder are conjugate diameter's of the curve of section: hence the line Op and the lines OPi lt 0/4 (which are the projections of the basal axes on the conjugate plane of Or), form a triad of conjugate diameters of the indicatrix, each being conjugate to the plane of the other two. This may also be proved analytically, as follows : Substituting the values of A\' y s^, x* y* z 2 ', given in Art. 9, we find that a XW + %/ 2fc' + cV* a r = LL -} a - 2 2 -} -*) W- a ) (>-i 2 The quantity within the brackets is zero, as may be seen from the last Article, or more directly by subtraction of the equations ~7 - h 4 *- 1 ( l 1 \ V* 2 ~ aV The coefficients of -g- 2 and 5 + ^ vanish, and the remaining teim ? 1/2 ?i r a may be transformed into THE OPTIC BI-KADIALS. 43 Ilence _W"^) W"aV (? *>)(' *V pZIWlZl lo , 2 ; U o There are four corresponding directions, namely, A/xi>, A/xv, A//,i/, \/AV, symmetrical to the principal planes of the indicatrix. 15. Given the direction of a line of transmission, to find the velocities of the corresponding rays. If RR' (Fig. 8) be parallel to the axis of the cylinder, and R' be the extremity of an axis of the base, it follows from Article 11 that R is a point on the indicatrix corresponding to a ray transmissible along the axis of the cylinder. The corresponding velocity being -^ = ^y>> it is seen that the velocities of transmission are inversely proportional to the axes of the base of the cylinder. An analytical solution is given in Art. 8. 16. The Optic bi-ra dials (secondary optic axes}. From Article 15 it follows that if the two velocities of transmission in a given direction are equal, the corresponding tangent cylinder has a circular base. But at every point K' (Fig. 8) on the edge of the base of such a cylinder, the normal of the basal section and therefore of the cylinder, and consequently also the normal of the cylinder and therefore of the indicatrix at every corresponding point K of the section conjugate to the ray, intersect the axis of the cylinder perpendicularly, and have the same length intercepted between the surface and the axis : hence every point on the conjugate section corresponds to a ray transmissible with the same velocity along the axis of the cylinder : the normals of the indicatrix at these points, and therefore the planes of polarisation, may have any azimuth whatever* 44 THE TRANSMISSION OF LIGHT IN CRYSTALS. That in the plane AGO there are two directions, and only two, namely those of the lines Os lf 0* 2 (Fig. 7c), for which the two velocities of trans- mission are equal, has already been proved (Art. 3). Along each of these lines Os^ Os. 2 ,Y&ys can thus be transmitted having any azimuth of plane of polarisation whatever, and the velocity of transmission is b for all of them : in the case of calcite and analogous crystals, such pro- perties only belong to that single direction which is termed the optic axis. By reason of this analogy, the directions Os^ Os. 2 , have been likewise termed optic axes. But not being perpendicular to the corresponding ray-fronts, they do not possess all the characters which belong to the optic axis of a uniaxal crystal : from another pair of directions, of which the optical characters are also such as in the case of a uniaxal crystal only belong to the optic axis, they have been distinguished as Secondary Optic Axes; and by Sir William Hamilton as Lines of Single Hay -Velocity. 1 In the case of a biaxal crystal, it is experimentally determined that none of the so-called optic axes, primary or secondary, have directions which pass permanently through the same lines of crystalline particles ; the lines of particles through which they pass differ with the colour of the light and the temperature of the crystal : hence the so-called optic axes have no material existence, and are in no proper sense of the word axes of the crystal. Where precision of thought and language is necessary, the lines may appropriately be termed the Optic Bi-radials, for they are directions in which a line is doubly a radius vector of the ray- surface : the term uni- radial has already been assigned a distinct signification by Mac Cullagh. 2 When the indicatrix is a spheroid at all temperatures of the crystal and for all colours of light, the bi-radial is found to be an axis of mor- phological and physical symmetry, and an axis of revolution of the ra}~- surface ; it always passes through the same line of crystalline particles : such a line may be regarded as a true axis of the crystal. 17. There cannot be more than one pair of optic U-radials. It has already been proved that Os lt Os 2 , are the only directions for which the velocities of the rays transmissible along the same line, lying in a plane of symmetry of the indicatrix, are equal : it remains to prove that there are no other bi-radials in any direction whatever. 1 Trans. Eoij. Irish Acad. : 1837, vol. 17, p. 132. 2 Ibid. : 1839, VoL 18, p. 40. THE OPTIC BI-BADIALS. 45 From Article 8 it is seen that the velocity of transmission r is connected with the values of X p v by the equation whence a*\* (r 2 - i 2 ) (r 2 - c 2 ) + iy (r 2 - 6- 2 ) (r 2 - a 2 ) + c V (r 2 - a ) (r 2 - & 2 ) = 0. Since a, b, c are in descending order of magnitude, the expression on the left-hand side of the last equation is positive, and therefore cannot be zero, if r has any value greater than a or less than c : hence no velocity of transmitted ray can be greater than a or less than c. Further, if fj. is distinct from zero, the above expression is necessarily negative when r = b; hence it changes sign and passes through a zero value as r decreases from a to I, and again as r decreases from b to c. If p is distinct from zero, the two values of r 2 which satisfy the above equation are thus unequal. Hence the bi-radials can only lie in the plane AOC. That in the plane AOC there are only two such lines may also be seen from the fact that for any direction lying in this plane one velocity of transmission is always b ; when the two velocities are equal the second velocity must also be b : hence if 8 is a point on the curve AC AC such that the perpendicular to the ladius vector conjugate to 08 is equal lo OB, the points S correspond to directions Os of single ray-velocity : there are four such points lying at the extremities of two diameters. The directions Os may also be readily found from the above general equation : for all rays lying in the plane A OC, p is zero, and the general relation becomes ** , ** -Q. **^~*> hence, the rays in this plane for which the two velocities of transmission are both equal to b are given by the equation which is identical with the equation given in Article 3. 18. Equation of the planes conjugate to the optic bi-radials. The equation of a plane conjugate to a line X^u v is For the bi-radials, ju=0 and i !L_ _n 1,2 ~2~T1.2 2 ~V, 46 THE TRANSMISSION OF LIGHT IN CRYSTALS. Hence their conjugate planes are given by the equation 19- The direction of a line Or being defined by its inclinations o- 15 o-.,, to the bi-radials Os lt Os. 2 , to find the jilanes of polarisation of the two rays which can be transmitted along il. Let [aj, [a a l, [r] (Fig. 9) be the sections of the indicatrix which are conjugate to the lines Oa lf 0a 2 , Or respectively, and let D ly D 2f be two adjacent points of intersection of [r] with [aj and [s 2 l. D! being common to the sections [r] and [aj , the tangent plane of the indicatrix at D l is parallel to both Or and 0a lt and therefore to the plane Orsj containing them. Similarly the tangent plane of the indicatrix at P 2 is parallel to the plane Ors 2 . FIG. y. Also, all planes tangent to the indicatiix at points on the sections [aj and fa 2 ] are equidistant from the origin (Art. 16) : hence the tangent planes of the indicatrix at Dj and D 2 are equidistant from the line Or, and are therefore equally inclined to the planes of polarisation, for the latter are the planes of symmetry of the elliptic cylinder which touches the ellipsoid in the section [r] (Art. 12). Hence the planes of polarisation of the two rays transmissible in the direction Or are the internal and external bisectors of the angle between the planes Ors lt Ors 2 . This is the first of the empirical laws of Biot (page 25). THE OPTIC BI-RADIALS. 4? 20. The direction of a line Or being defined by its inclinations jind d are adjacent points on the ray- surface, and the line rd is thus tangent to the rav- snrfacc at r. The tangent plane to the ray-surface at r is the plane of the ray-front corresponding to the ray Or : it must pass through all lines tangent to the ray-surface at r, and thus through the line rd, and be perpendi- cular therefore to the plane ItXOr. FIG. 10. (/;.) The plane RNOr (Fig. 11) also will intersect the indicatrix in an ellipse : let G be a point on this ellipse distant from R by an arc which is a small quantity of the first order : to this order of small quanti- ties GH, the normal of the ellipse at (r, is also the normal of the indicatrix at that point. Hence if ffOH be perpendicular to GH and % = /r> 0(j is the direction of the ray corresponding to the point (7, and y is a point on the ray- surface : in the same way as before it follows that ry is a tangent line of the ray- surface, and is thus the intersection of the tangent plane at r with the plane RNOr. Let Or, Oy, intersect the ellipse in the points R', G', respectively : then OR' and OG' are respectively conjugate to OR and OG, being perpen- dicular to RX and GH, and therefore parallel to the tangents at R and G : the area of a parallelogram of which the adjacent sides are conjugate radii vectores is constant ; hence Also, by construction, hence Or: OR' = Og: OG', 50 THE TBANSMISSION OF LIGHT IN CKYSTALS, and the line rg is therefore parallel to the tangent of the ellipse at IT, and consequently to the line OR which is conjugate to OR'. Hence the ray-front corresponding to the ray Or intersects the trans - Terse plane RNOr perpendicularly in a line parallel to OR. The diametral line Of, perpendicular to OR and lying in the plane RXOr, is therefore normal to the ray-front corresponding to the ray Or (Fig. 12). FIG. 11. Analytical Proof. The following is interesting to the mathematical student, by reason of the eliminations: From Article 7 we have i / hence * Remembering that ^ + A? + ? = 1 (Art. 7), we have xx'+y-y'+zz' A. (2). Also tPxtf + V*yy'+<*zz' =0 (Art. 7). (3). It is thus required to determine the tangent plane at a point xyv of the ray-surface in terms of the co-ordinates x'y'z', which are connected by the above equations and also by the relation cV 2 =1. (4). THE RAY- FRONT CORRESPONDING TO A GIVEN RAY. 51 Forming the differential of each of the equations (1), we have -a a )5.j;' + 2r.v'5r = Alx+xlA } Multiply these equations by V, &y, cV, respectively, and add: the quantity 6^4 is thus eliminated, for its coefficient a?.i\v'+b*y vanishes hy relation (3) ; we then have (> _ a *)tf y . t .' 4. ( r 2 _ l*)&y'frj' + (r 2 - c 2 )cV6s' -f 2rcV = Remembering that a*jfbjf+!tyfy+Af&*f = 0, owing to relation (4), we have - (aV&p' + i 4 //' %' + c 4 V^') -f 2rSr = J (aV5.t- + %'% + c^'Si). But by Article 4 aV 2 + ^// 2 + ^' 3 = r !l ; whence 4s/M +$&+&& rgr. Substituting this value in the preceding equation, we have rbr = ^(a or (a? - 4aV)g.i> + (>j If Z ??& ?i be the direction-cosines of Of, the perpendicular to the tangent plane of the ray-surface at x y z, we must have l&v + mfy + nSz = 0, whence (a.) Each of these quantities is equal to Ix'+my'+nz' la' -\-rny' + The denominator of this expression is zero, by relation (2) ; hence the numerator /.;' -\-m)/' -\-m' is also zero, for the three equivalent fractions are never all of them indeterminate, and are none of them infinite. From the relation lv' -\-rny' +nz' = 0, (7). it follows that Of is perpendicular to OR. (b.) Also, multiplying both numerator and denominator of each of the fractions (6) by y'z\b*-c*), sV(c 2 -a 2 ), x'y'(a?-b*), respectively, we find that each of them is equal to _ ly'z' (ft - c 2 ) + mz'x' (c 2 - a 2 ) + nx'y' (a 2 - //-Q _ y'z' (6 2 - c 2 ) (x - A () (2 - Ac*z') ' 52 THE TRANSMISSION OF LIGHT IN CRYSTALS. On expanding the denominator, it will be found that the terms involving A mutually destroy each other, owing to the identity a *(l* - c*) + 6V - a*) + cV - 6 s ) - : the denominator thus reduces to ' f-a zx'' V or owing to the equations (1). When multiplied out, this term is likewise found to be zero. Hence the numerator of the above expression is also zero, and we have the relation ly'z'(b z - c*) +m*V(tf* - a 2 ) +nafy'(

/ is the foot of the perpendicular drawn from () to the tangent plane of the ray- surface at r, and Of is the resolved velocity of the ray Or along the normal to the corresponding front. But Or, Of, are by construction perpendicular respectively to RN and UN Of RO : hence the triangles rfO, ONE, are similar, and 'Tr> == /T"' Also, by construction, HL\ = -Q~; hence Qf == 7j-R t FIG. 12. 24:. The line OR is always a normal of the curve in which the indica- trix is intersected by a central plane parallel to that ray-front ivhich cor- responds to the ray Or : in the general case, OR is an axis of the curve. RN (Fig. 12) being the normal of the indicatrix at R, any line per- pendicular to RN and to the plane RNOr is tangent both to the in- dicatrix, and to the section of the indicatrix made by any plane which is perpendicular to the plane RNOr at the point R ; it is thus tangent to the particular section made by that plane of the series which passes through 0. Of this section OR is a central radius vector : hence the tangent at R to the section is at right angles to a central radius vector. The section being in general an ellipse, R is in such case the extremity of an axis of the section. Hence it is seen that the ray- surface is the envelope of planes which are distant from a parallel central section of the indicatrix by the inverse lengths of the semi-axes of the latter curve : which is virtually Fresnel's geometrical construction of the surface. Conversely, 6-1 HIE tUAXSJIlSSlUN OF LIGHT IJJ CRYSTALS. 25. If OR iv a central normal (and therefore in general an axis) of the curve in which the indicatrix is intersected by a plane parallel to a given direction of jay-front, the plane through OB normal to the direction of the ray-front contains the ray Or, which corresponds to the point li, and also the line luY, which u the normal of the plane of polaiisation of tlis nty. The radius vector Oil (Fig. 12) bein;? a central normal of the curve of intersection, aline perpendicular to 012 and lying in the plane parallel to the ray-front, is a tangent to the curve of intersection at 11 : hence 12A' the normal of the indicatrix at R must lie somewhere or other in the plane ROf perpendicular to this line. And the ray Or must tie in the same plane. FIG. 13. 26. The two rays corresponding* to a given direction of front-normal. Hence if only the direction of a ray -front be given, there are in general two corresponding positions of the ray-front, or, in other words, of tangent planes to the ray- surf ace : and for each there is a corresponding ray (Fig. 13). The rays lie each of them in a plane containing the central normal Of and one of the axes 012, 07', of the section of the indicatrix by a plane parallel to the ray-front ; they are thus in two perpendicular planes which inter- sect in the line Of : the corresponding velocities resolved along the given front-normal are measured by jj^ and ^ respectively : the normal of the plane of polarisation is parallel to RN for the ray Or, and to TX' for the ray 01. The directions of vibration at points of the respective rays, ac- CORRESPONDING RAYS AND FRONT -NORMAIxS. 55 cording to the latest version of the elastic theory (Art, 1), are thus indicated by the shading in Figure 13. It may be remarked that the planes of polarisation of the rays Or, Ot t though perpendicular to the normals RN, TN', are not perpendicular to each other ; for it is the lines RO, TO, not the lines RN, TN', which are at right angles : it is easily seen that the cosine of the angle between the planes^ is equal to sin fOr sinfOt. Hence only the transverse planes, not the planes of polarisation of the two rays, are perpendicular to each other. 27. The two front- normals corresponding to a given direction of ray. Similarly, if only the direction of a ray be given, there are in general two corresponding positions and directions of the ray-front, and two corres- ponding rays (Fig. 14). The front-normals lie each of them in a plane FIG. 14 containing the ray-direction and one of the lines OE^ OR 2 ; they are thus in two perpendicular planes which intersect in the line Or : the correspond- ing ray-velocities are measured by -- , , respectively : the normal of the plane of polarisation is parallel to R^ for the ray Or lt and to R 2 Nz for the ray Or z . The directions of vibration at points of the re- spective rays, according to the latest version of the elastic theory (Art. 1), are thus indicated by the shading in Figure 14. 28. Given the co-ordinates x'y'z' of R, to find 0, the angle between the corresponding ray Or and its front-normal Of. Or, Of, being perpendicular toRN, RO, respectively (Fig* 12), 7?iy -i cos = cos 1 0f= cos NRO = = -_. G THE TRANSMISSION OF LIGHT IN CRYSTALS. hence taird = / V' 2 1 29. Crivew i/i6 direction-cosines X p v of a line of transmission, to find 0, the angle between the corresponding ray Or and its front-normal Of. Find /Y 2 , r. 2 2 , (Art. 8), and then .r////^', -^V^', the co-ordinates of the points #!, 7? 2 (Art. 9) I also y/ 2 = .' 1 ; ' 3 +// 1 '' 2 + ^i" and r/ 2 = .c./ 2 + /// 2 + V 2 ; we have sec 6 { = TV/ ; sec 0. 2 = r. 2 r./. 30. // a ray lie* in a yivcn axial 'plane oj the indicatn'.c, to find the direMmsfor which tli3 i.idiiuitivn ti tj the front-normal is a iii'^'mmii. First method. Lot the given axial plane be AUG. Each direction of transmission lying in this plane corresponds to two points 7Z 15 E^ on the indicatrix : one of these, 7? 2 , always coincides with B, and the corresponding ray coincides with its front -normal; the other, /i 1} is in the plane AOC, and the corres- ponding ray coincides with its front-normal only when E l is at A, A,C, or C. If ,v' z' be the co-ordinates of 1'n, tan 6=(c 2 -(t?) z'*v' (Art. 28). Hence, writing e'V* = 1 V' J , we have For a maximum value of 6, tt'V 2 = i = r i .c'-. '>r being parallel to the tangent of the indicatrix at 7^, tan rUA = -'','', - c~; e or st method. Taking AOC for the given axial plane, as in the preceding Article, we have, when aV 2 ^, (I C and cot 61 = tan 2 AGO, or 0=---- 2 A CO. a Hence the maximum or minimum angle which a ray lying in the axial plane AOC can make with its front is given by the angle AC A. Second method. This result is also manifest from the fact that when the ray Or is paral- lel to AC, the conjugate diameter Oli lt and therefore also the ray-front, is parallel to CA ; as in the second proof given in Art. 30. 32. Given the co-ordinates x'y'z' of R, to find the direction-cosines I m n of the normal Of to the corresponding ray-front. For any line Imn in the plane RNOr, as already proved in Article 6, we have the equation ly'z' (b* - c 2 ) + w/,5 V (c 2 - a?} + nay (a 2 - 6' 2 )=0. If the line linn is likewise perpendicular to OR, of which the direc- .*/ /' c' lion-cosines are -,- V~T' we have also r r r From these equations the ratios I : m: n are found to be : I m n D = E = F where D=tfz'*(i*-a*)-x'y'* (a 2 -b*) = x'(l -aV a ), E=y\v'* ( 9 -fc a ) -7/V 2 (b* -c 3 ) =y f (l -6V a ), ir= c ^'* (>-c a ) - 5V a (c a -a a )=5'(l -cV a ) ; whence / m n .r' (1-oV 2 ) V (1 JV 2 ) -' (l-cV' a )' where r'^^yt^.r^ + f/^ + c" 2 . These equations determine tlic direction-cosines / ;// n of the normal of the ray-front corresponding to the point .r'y'z' or R. 58 THE TRANSMISSION OF LIGHT IN CRYSTALS. 33. Given the direction-cosines I mn of the f font-normal Of, to find f it f. 2 , the respective velocities of the tw^ corresponding ray -fronts resolved normally to them. In Article 23 it was shown that the velocity of the ray-front of the ray Or resolved normally to the front is ,: denoting the resolved velocity by / and substituting ' f= = in the equations of the last Article, we have I m n Each of these fractions is equal to I m '72 la 4" m 7z But the denominator of the last expression has been proved to be zero (Art. 32) ; hence the numerator is also zero, for the fractions equivalent to the expression are never all of them indeterminate and are never infinite ; we thus have This is a quadratic equation in/ 2 , and its two roots fi, / 2 2 , are the re solved velocities required. Multiplied out it takes the form p _/2 .[ 34. Given /i, / 2 , the velocities of normal- transmission of two ray fronts having a common direction of normal 0/i/ 2 , to find I m n, the direc- tion-cosines of the latter. From Art. 33, Z 2 m* n z 9 -* 8 -A " /a Determining the ratios I 2 : ui 2 : n 2 from these equations, we find that RAY-FRONTS AND CORRESPONDING POINTS ON THE INDICATEIX. 59 The sum of the numerators of the fractions is unity : the sum of the denominators is 6" - a 2 ) + 6- 4 (a 8 - Z/ 2 ). The coafficients of /i 2 // and fi~\-f* vanish, and the remaining term may be transformed into (a 2 - c 2 ) (6 2 - c 2 ) ' There are four corresponding directions, namely, Imn, Imn, Imn, Imn, symmetrical to the principal planes of the indicatrix. 35. Given the direction-cosines I m n of the normals 0/j, Of 2 , of two ray-fronts having the same direction, to find the co-ordinates x^y-^z^, t T 2 'f/ 2 'c./, of the corresponding points H, T, on the indicatrix. The values /i 2 ,/ 2 2 , having been found by the equation of Article 33, the co-ordinates x y /, x 2 ' y% z% of the points E and T respectively are determined by the following equations, also from Art. 33 : 2Al = _*iL . n /I*-* 8 f -2 -J / -1 12 /' 2 ,,2 /a ft /t * /a - remembering, also, that the co-ordinates of each point must satisfy the equation of the indicatrix. It will be observed that the above equations are identical in form with those given in Art. 9 : in the one case the direction-cosines and velocities are those belonging to the rays, and in the other case are those belonging to the front-normals. 36. Given f n f 2, the velocities ff normal- transmission of ttvo ray -fronts having a common direction of normal 0/i/ 2 , to find the co-ordinates x-^y^^ r a'//a'~a' f ^ l& corresponding points R, '1\ on the indicatrix. From Art. 35 n A 00 THE TRANSMISSION OF LIGHT IN CRYSTALS. The sjaare of each of these fractions is equal to 7 . '2 _1_ ,, >-2 I ^ 1-2 *! T }/i 1-1 Hence, reuaembering that .r/ 2 -f /i' 2 + V 2 == OR 2 = 77, > we find on Ji" substituting the values of I 2 , w 2 , n' 2 , given in Art. 34, ff -'~(*-*)(*-V)W'-'*) 2 ~ /A flW, all vanish, and that the coefficients of /^and// are equal but of opposite sign : the numerator then takes the form ( A 2 ~/2 2 ) (a a - i 2 ) (A a - e 2 ) (c- 2 - 2 ) Hencp ^2 _ /i 2 (/i 2 ~/2 2 ) _ (ff-? ff (f? -/,) ( C 2 - 2 ) ( 2 - i 2 ) ' Corresponding expressions give the values of i// 8 , %' 2 , ar a fa , 2/ 2 ' 2 > ^2 f2 - The above relation, with many others of this Chapter, was first given by Prof. Sylvester, 1 starting from the vibrational inferences of Fresnel. 37. Given the direction-cosines I m n of a front -normal Of, to find those of the corresponding rays Or, Ot. Find the co-ordinates of R and T by the method of Art. 35, and then the direction-cosines of the ra} r s Or, Ot, by means of the equations in Article 6. 38. Given the direction-cosines of a ray Or, to find those of the cor- responding front-normals. Find the co-ordinates of 1^ and R 2 (Art. 9), and then the direction- cosines of the corresponding front-normals by Article 32. 39. The front-normal surface, or pedal of the ray-surface. It was shown in Art. 33 that if / be the velocity of transmission of a ray-front resolved along its normal Of, of which the direction-cosines arc / in n, 1 Philos. Magazine, ser. 3 : 1837, vol. 11, pp. 461, 537; 1838, vol. 12, pp. 73, 34 1. SURFACE OF WAVE-SLOWNESS Oil INDEX-SURFACE. Gl Hence, if .r // ^ bo the co-ordinates of /, and tlio length of 0/be de- noted by r, we have r == f'. -v = lr, >J == ni>'> ~ iir* Substituting in the above equation, we get This is the equation of the locus of the points /, or of the pedal of the ray-surface : the velocity of normal-propagation of a ray-front along any radius vector of the surface is measured by the length of the radius vector. 40. Ihe polar reciprocal of the ray-surface belongs to the seme family ; surface of wave-slowness or index-surface. The radius of a concentric reciprocating sphere being taken as unity, the pole 77 which corresponds to the ray-front will lie in the front-normal Of at a distance ^ from the origin. Hence if xyzr refer to the point/, and ??<;> to the pole of the ray- front, we have - _ .1 !L -. - - : j = -- r~ p' r~~ p' r ~ p p Substituting these values in the equation of the locus of the points /, \vo find for the equation of the polar reciprocal of the ray surface 01 This is a surface of the same family as thcTay-surfacc, being derived .r" i/" c" from the ellipsoid 4- ' 4- = 1 in the same way that the ray-surface O \J or Since a, b, c are in descending order of magnitude, the expression en the left-hand side of the last equation is positive, and therefore cannot be zero, if / has any value greater than a or less than c; as is otherwise evident from the fact that /= ^, where R is a point on the indicatrix: hence no value of /greater than a or less than c can make the expression zero. 61 THE TRANSMISSION OF LIGHT IN CRYSTALS. Further, if in is distinct from zero, the above expression is necessarily negative when /= I : hence it changes sign and passes through a zero value as / decreases from a to b, and again as / decreases from b to c. If m is distinct from zero, the two values of J' 2 which satisfy the above equation are thus unequal. Hence the bi-normals can only lie in the plane AGO. Since OB is normal to the plane of polarisation for any ray-front of which the normal lies in the plane AOC, one root of the equation corresponding to such a ray-front is always /=J, and this must be the value of the equal roots : the directions of the bi-normals may therefore be foucd directly from the general equation (Art. 33) as follows : For any front-normal in the plane AOC, m = Q, and the values of / 2 are l z n 2 given by the equation 2 _ 2 + TaT^a = : hence, the directions of the front-normals in the plane AOC for which f-b are given by the equation I 2 n 2 - 2 = r^ 2 d" u C ; which is identical with the equation of last Article. 43. The direction of a line Of being defined by its inclinations ir it ir z to the li-normals Op lt Op z , to find the transverse planes of the two rays of which the corresponding fronts are perpendicular to the given line. Let Q)J , [p. 2 ] , be the circular sections of the indicatrix perpendicular to the bi-normals Op 1} Op. 2 , respectively, and let [/] be the central section of the indicatrix parallel to the given direction of ray-front (Fig. 16). Let [/] intersect [p^, [p 2 ], in E lt E 2 , respectively. All radii vectores in the two circular sections being equal, OE l OE. 2 : and the axes OR, OT, of the elliptical section [/] are therefore the in- ternal and external bisectors of the angle E^OE^. By Art. 26 the two rays Or, Ot, corresponding to the front-normal of are in the planes fOR, fOT, respectively. Again, Of is perpendicular to both OE 1 and OE 2 ; Opu Op. 2 , are perpendicular to OE^ and OE. 2 respectively. Hence OE 1 is perpendicular to both Of and Op lt and therefore to their plane /Op^\ OE Z is perpendicular to both O/and Op. 2 , and therefore to their piano ft- The planes fOE : then Sea = MOfj. = mOSi = 0. Let the angle ns^i be denoted by 6 : to determine the relation between ns 1 and the angle 0, we thus require to express ON or Se in terms of the angle ns^n or aOjj.. We have Sv = ye tan = Oo- cos tan y - cos 6 tan 0. Hence n^ = b*-ON = tf'So- = b cos 6 tan = s^m cos 0. The angle mn*i is thus a right angle ; and the locus of n is a circle passing through the point s it and having w^ for diameter. Second proof. Let x'y'z' be the co-ordinates of any point S on the section conjugate to the bi-radial Os 1 : by Article 18 Now 2/', being the perpendicular from S or o- to the plane MfjtM's lt is equal to o-0sinpOn, a line perpendicular to the plane which is conjugate to that bi-radial ; a rela- tion by means of which the above value may likewise be obtained. 48. Polarisation of the ray corresponding to a given front-normal of the bi-radial cone. For the ray transmissible along Os l which has On for its front-normal, the normal of the plane of polarisation is SX or ns l : hence the plane of polarisation of that ray Os 1 which has On for its front-normal, meets the base of the cone in a line parallel to the line nm, or in other words in the line which joins Sj to the other extremity of that diameter of the circle which passes through n. 49. The bi-normal cone (the cone of rays corresponding to a front which is perpendicular to a bi-normal). In general (Article 26), if OR, OT, (Figs. 13, 18) be the axes of a central section of the indicatrix, the points R, T t correspond to rays Or, Ot, having fronts in the same direction, namely parallel to the plane OUT : also, if Of be the normal to the fronts, the rays Or, Ot, lie in the planes fOR, /Or, and are perpendicular to the lines which are normal to the indicatrix THE BI-NOKMAL CONE. GO at 7t and T respectively. But we have seen that all points on the circular section perpendicular to a bi-normal 0^ correspond to rays having the same position and direction of front, the latter being parallel to the circular section : further Op l is not an axis of the indicatrix, and thus is only coincident with the corresponding ray for the two points B, 13, on the circular section . Hence, as the point R moves round the circular section of the indica- trix, the ray Or, which is always in the plane R0p 1} describes a cone of which the bi-normal Op 1 is an edge, for it corresponds to the points B, B t on the curve. The cone may be conveniently designated a bi-normal cone. Corollary. Since every front touches the ray-surface where the cor- responding ray meets it, a bi-normal is perpendicular to a plane which touches the ray- surface in a closed curve. In the next Article it will be shown that this curve is a circle. w. FIG. 18. 50. A plane perpendicular to a bi-normal intersects the cone of cor- responding rays in a circle. Let TF, W, (Fig. 18) be the points where the circular section inter- sects the plane AOG : let 0^ = 3, and R be any point on the circular section : the plane R0p lt containing the ray Or corresponding to the point R, will intersect a plane, drawn through p^ parallel to the circular section, in a line p^r parallel to OR ; similarly, if Ow be the ray corres- ponding to the point W t p^w is parallel to OW : hence the angle rp^w = angle HOW. Penote it by 0. Also +jy a +*V a -i a , if fltyV be the co-ordinates of the point 11 (Article 4). 70 THE TRANSMISSION OF LIGHT IN CRYSTALS. But y' being tho perpendicular from E on the plane TFO//J or AOC, we have y'=OR sin TFOB = -5|1. Also, since 7i is on a circular section, Finding a;' and z' in terms of by means of this equation and the relation aV+ay a +cV a = 1, we get j ft a - c 2 a a 2 - b' 2 co*8 Substituting these values of a;' 2 , i/' 3 , s' 3 in the equation for p- we have lh r = *--? /s of the bi-nrrmal cone. The plane of polarisation of the ray Or is a plane perpendicular to the transverse plane Op^r. The line Op lt being perpendicular to the plane p^rii', is perpendicular to the line joining p 1 to r, the other extremity of that diameter of the circle p^'ic which passes through r ; the line rp lt being perpendicular to both p^O and /v'> is perpendicular to the plane O/^r containing them: as any plane passing through /v' or its parallel nc, is likewise perpendicular to the plane Opf, the plane One is the plane of polarisation of the ray Or. DERIVATIVE SURFACES. 71 53. Representative surfaces derived from the Indi- catrix. (a) The characters of a ray of light transmitted in a crystal may also be expressed by reference to corresponding points on the polar reciprocal of the indicatrix relative to a concentric sphere : this surface is an ellipsoid represented by the equation 2 + ^ + =1 (p. 105 and Fig. 19). If OR, a radius vector of the indicatrix, be normal to a tangent plane of the polar reciprocal of the indicatrix, meeting the plane in a point M, OR'OM=l t if the radius of the reciprocating sphere be unity. If P be the point in which the tangent plane perpendicular to OR touches the polar reciprocal, and PG be the normal of the latter surface at the point P, the lines PG, PO, lie in the plane RNOr : let PG intersect the ray Or in the point G. If m be the point in which OP intersects the plane which touches the indicatrix at R, OP-Om = l, and OP is thus the inverse of RN. Hence, to every point P on the polar reciprocal of the indicatrix corresponds a ray Or : it lies in the plane PGOr, and is per- pendicular to OP : its velocity of transmission is measured by OP : its transverse plane is PGOr : the ray-front intersects the transverse plane PGOr perpendicularly in a line parallel to PG , and its velocity of normal- transmission is measured by PG. (b) Von Lang 1 has pointed out that if a surface be derived from the ellipsoid aV-j-6V-j-cV=l by elongating each radius vector until the new length is measured by the nth power of its original value, the derivative surface may likewise be used for the geometrical representation of the characters of transmitted rays. This result can be generalised still farther, as follows : Let $~ l (r) be any function of r, which always increases and decreases with r, or vice versa : it will have an apsidal (i.e. maximum or minimum) value at the same time as r. If then a new surface be derived by elongating each radius vector r of the indicatrix to a length p, determined by the relation p = ^~" 1 ( / ') or r = ( i>(p)> a central section of the new surface will have its apsidal diameters in exactly the same directions as those of the section of the indicatrix by the same plane. If p it p 2 , be the half- lengths of the new diameters, the corresponding ray-fronts are respectively at distances - and - or rr - and r7 - T from the central section ; the ray- surface itself is the envelope of these planes. 1 Sitz. Ak. Wien, 1861, vol. 43, sec. 2, p. 645. 72 THE TRANSMISSION OF LIGHT IN CRYSTALS. The general equation of the new surface is easily found : If r be the length o f a radius vector of the indicatrix and I m n be its direction-cosines, a 2 / 2 + IV +cV =- ; 17 4 being the co-ordinates of r 1 the corresponding point on the new surface, f = fy, ?? = wfp> 4 = /? : whence a f 2 + #V + c 2 4 2 = or 'F + V + ^ which is the required equation. Fresnel's " surface of elasticity" is the particular case in which (p)=-, for the equation then becomes fl a ^+ #y + c* 2 = ( + /r + T 2 )' 2 - For the ** surface of elasticity," the transverse planes of the rays corres- ponding to a given direction of ray-front pass through the apsidal dia- meters of a central section, as in the case of the indicatrix, but the distance of the ray-front corresponding to a semi-diameter of length p l is not as in the indicatrix, but; - or p^ Pi 0(/>i) The corresponding ray is only perpendicular to the corresponding normal of the representative surface in the case of the indicatrix : in every case, however, the normal of the representative surface lies in the plane passing through the corresponding diameter and the front-normal : for the curves of intersection of the two surfaces by the given plane have parallel tangents at the extremities of their maximum and minimum diameters. Hence, as in the case of the indicatrix, the plane passing through a diameter and a normal of the surface at the extremity of the diameter is the transverse plane of the corresponding ray. (c) In exactly the same way a series of surfaces can be derived from the polar reciprocal of the indicatrix. The above generalisation serves as a reminder that there is not neces- sarily a simple relation between a surface of geometrical representation and the characters of the ether. 73 CHAPTER V. VARIOUS OPTICAL RELATIONS WHICH ARE INDEPENDENT OF THE PHYSICAL CHARACTER OF THE PERIODIC CHANGE. 1. In Chapter II we have shown that after the discovery of the polarisation of light by reflection by Mains in 1808, and of the corres- pondence of optical and morphological symmetry by Brewster in 1819, the true laws of transmission of light in biaxal crystals must soon have been suggested, independently of any hypothesis as to the physical character of the periodic change : in fact, their enunciation by Fresnel in 1821 was only two years later than Biot's discovery of two empirical laws by which the accuracy of a geometrical representation could be tested. If the truth of the construction given by Huygens for the case of calcite is acknowledged, the suggestion presents itself as soon as the planes of polarisation of the two rays transmissible in any direction in a crystal of calcite are represented by their normals. In the present Chapter we proceed to indicate very briefly, for the convenience of the student, various other important relation s ; which, though really independent of any hypothesis as to the nature of the periodic change, are usually imagined and expressed as belonging to an elastic ether. It will at the same time be shown that the form of the ray- surface for biaxal crystals is not merely suggested by a geometrical generalisation as a tentative one, but is a necessary consequence of the difference of symmetrical development of the same physical characters, whatever they may be, which originate the sphere and spheroid of a uniaxal crystal : it will further be shown that the same form of the ray- sut face would result from the general features of perpendicularly transverse vibrations, and be independent of the real nature of the periodic change. Preliminary algebraical expression for the transmission of a ray of common light. 2. It will be convenient, in the first place, to find a mathematical expression connecting the magnitude of the disturbance or change of state at any point in a ray of common light of simple colour with the position of the point, the time, and the period of the vibration. For this purpose it is necessary to make an assumption as to the law of the change : the simplest which can be made is that, at any point of a ray of common light y = asm f* THE TKANSMISSION OF LIGHT IN CKYSTALS. of simple colour, the variation of the state with the time follows the same law as the variation of position of an isochronous pendulum. It will be found that, for a ray of simple colour, the expression Say . . \^ is one which satisfies this condition and is consistent with all experi- ments as yet referred to ; x denoting the distance of any point in the ray from a fixed point in it, y the magnitude of the disturbance or change of state at the point x at the time t ; v the velocity of transmission, X the wave-length, a and a two constants for all values of x and t : 1. At a given point, indicated by its distance x from the origin, the change of state varies periodically with the time t : the same value of y, and therefore the same change of state, recurs whenever the expression 27T -JT (I'tay-fca increases by 2?r, that is when t increases by the constant interval . The same change of state recurs, but with opposite sign, whenever the expression ~(vt o?) + a increases by TT, that is when t X increases by half the above interval. 2. At a given instant, indicated by the time t, the change of state is the same in magnitude and sign for all points separated from each other by the distance X : it is the same in magnitude and opposite in sign for all points separated from each other by half that distance. 3. The relation between y and t is identical with the relation between the position of an isochronous pendulum and the time. a, being the maximum value of y, is the amplitude of the vibration. & (vt CT) + CI being the phase of the vibration at the point ,v at the X time t, a is the phase of the vibration at the origin (x = 0) at the epoch from which the time is measured ( = 0). The period of the vibration being independent of the amplitude, the law is consistent with the independence of colour and intensity. Conversely, if the magnitude of the change of state at each point of a line is given by the expression y asm\ ~-(i't~*- .)-}- a r, and the change 1 * j is of the physical character which belongs to light, a ray of light of simple PRELIMINARY REPRESENTATION OF THE PERIODIC CHANGE. 7^ colour is passing along the line with a velocity v : the intensity corres- ponds to the amplitude a, the colour to the period , while the phase of the vibration at the origin at the initial epoch is a. From analogy with sound, we may tentatively assume that the intensity of the light corresponding to this simple change of state is measured by the square of the amplitude. Resultant effect of the simultaneous transmission of two or more such rays along the same line. 3. The fact of the periodicity of the change was deduced from experiments relative to the mutual interference of rays of light : it is easily seen that the above expression for the change, combined with the principle of superposition, is consistent with the observed phenomena from which it was deduced. (a) If the component rays have the same wave-lenrjlh and velocity. 1 . For let two rays of the same simple colour, transmissible along a given line with the same velocity v, be represented respectively by the expressions = rtsin| -~ (ttf-.i-)-f a \ if bjth are transmitted simultaneously, the principle of superposition requires the resultant change to be determined by the expression y = a sn (^-.r) + | +6 sin j ^(rt- If the terms can be addel together in the swie way as numerical cmtiiie.i of a sinyle binl, !/ = (a cos a + 1) cos /3) tin -~ (vt ,r) + (a sin a-f b sin ft) cos y (i:t - .v) if c 2 = a 3 + i* + 2ab cos (a - ft) a sin a + b sin /3 and tan y .-, ------- n . a cos a + ycos p Hence the resultant effect of the t\vo rays is identical with that of a single ray transmitted along the same line with the same velocity and the same wave-length (and thus of the same colour), but having an intensity r and an original phase 7. And the intensity of the resultant ray depends 76 THE TRANSMISSION OF LIGHT IN CRYSTALS. not only on the intensities of the component rays but on their difference of phase at the same point at the same instant : if a and b are equal and a differs from /3 by any odd multiple of TT, the intensit} 7 of the single resultant ray is constantly zero. 2. It may in this way be shown that the resultant effect of the simul- taneous transmission of any number of such rays of the same simple colour along the same line with the same velocity is identical with that of a single ray of the same colour and velocity, and having a determinate phase and intensity. (b) If the component rays differ in icave-lenatli or velocity. On the other hand, if the component rays differ either in velocity or wave-length, the resultant effect is not that of a single ray of simple colour : the resultant effect is still expressed by a sin j (vt .r) -j- a ] -j- b s (2jr but the expression cannot take the simpler form csin^ tt ( X ) in which c and y are both constants : indeed, the resultant effect is not peiiodic at all unless the ratio: is commensurable. A A Kinematical representation of the periodic change at any point of&ucli a ray. 4. Whatever be the physical character of the periodic change at any point of a ray of light, the state at any point P at a given instant may thus (consistently with any facts as yet indicated) be represented by the above expression this algebraical expression may in turn be represented geometrically ; the magnitude y being represented by the distance of a point p from the point P, and the distance being considered positive or negative according to the direction in which it is measured. The phenomena of interference, from which the above expression has been deduced, merely require the direction in which the line Pp is measured to be necessarily the same for all points of the same ray, and for all interfering rays transmitted along the same line. This mode of representation in no way assumes that the actual change of state at the point P is a to-and-fro motion of a particle of ether in the arbitrary Hue Pp ; the direction of the line Pp is required to be constant merely to secure that the changes, if they have any directional character MORE GENERAL REPRESENTATION. 77 at all, ma}' be added together like simple numerical quantities of the same kind: that the change is really directional in character may be in- ferred from the fact that it is being transmitted in a definite direction through the medium. In exactly the same way, the transmission of a ray of light along a line is sometimes conveniently represented (in the discus- sion of aberration, for instance) by the transmission of a point along the line with constant velocity, although light is certainly not due to the transmission of a particle along the direction of the ray. Preliminary algebraical expression for the transmission of a ray of plane-polarised light. 5. It was found by Fresnel, in conjunction with Arago, that two rays of plane-polarised light, if their planes of polarisation are parallel, mat/ mutually interfere in exactly the same way as ordinary light : hence, as far as this experiment goes, the periodic change at any point of a plane- polarised ray can be represented in exactly the same way as for common light ; the only difference being that while a common ray is so far analo- gous to a circular cylinder that its characters are identical on all its sides, a plane-polarised ray is analogous to an elliptical cylinder to the extent that the properties of the ray are dissimilarly symmetrical relative to two perpendicular planes (pages 12 and 17). If all the characters of a plane-polarised ray can be accounted for by such a kinematical representation as is mentioned above, the line Pp must lie either in the plane of polarisation or the transverse plane ; but it may have any inclination whatsoever to the ray, so long as for two interfering rays the direction is identical. More general representation of the peiiudic change at any point of a common or plane-polarised i\,y. 6- Since> as far as the above experiments are concerned, the inclina- tion of the direction Pp to the line of transmission of either a plane- polarised or a common ray, may be any whatsoever, it follows that the change may really not be simple, but multiple in direction ; assuming that each transmitted periodic change will interfere for itself, as if those having other directions did not exist. In fact, it will be seen that the periodic change may likewise be repre- sented by the composite expression 7r _ - consisting of any number of terms : for each separate term, independently 78 THE TRANSMISSION OF LIGHT IN CRYSTALS. of its directional relations, resumes its original value at distances along the ray separated from each other by the common length A, or at times sepa- rated from each other by the common period : hence, if two rays an- 10 nihilate each other under given circumstances, annihilation will again take place if one of the rays is moved parallel to itself through the distance \ along its line of transmission. And it is important to remark that as each term recurs individually after the same interval of time or distance, the whole expression likewise recurs and has the same total value, even if the terms are not subject to the same law of addition as simple numerical quantities. It will also be obvious on reflection that any ray which is within the reach of experiment is necessarily composite as regards the origin of its vibration, even if it be simple as regards its colour: the luminous source is not a geometrical point, but a surface of considerable dimensions as com- pared with the wave-length of a ray of light ; hence the periodic change, of which the effects are observed at a given point of a line of transmission, is really of composite origin and due to the superposition of the periodic changes transmitted from the points of a luminous area of appreciable magnitude. As for the difference between a common and a plane-polarised ra} T , the first suggestion w r hich presents itself is that the latter is due to the dis- tortion of the common ray from which it was derived ; just as an ellip-, tical cylinder may be derived from a circular cylinder by compression in a direction inclined to the axis. Experimental discovery made ly Fresnel and Arago. 7. (a) But Fresnel and Arago found that, when one of two interfering plane -polarised rays is turned through a right angle round its direction of transmission, the interference -effects completely disappear, whatever the difference of phase of the two rays. Hence, with this relative position of the planes of polarisation, the periodic change produced at any point by the transmission of one ray is in no direction coincident with a periodic change produced by the transmission of the other ray ; for as we have seen (Art. 3), such coincidence would involve a variation of intensity of the resultant effect : if this be granted, it follows tbatfor a plane-polarised ray the actual periodic change must be in only a single direction, and the single direction must be perpendicular to the line of transmission ; for otherwise the two positions of the plane-polarised ray would give two positions of the periodic change which would have a resolved part in DISCOVERY MADE BY FRESNEL AND ARAGO. ' 79 common. Since the direction is single, it must be in one of the sym- metral planes of the ray : hence the direction of the actual periodic change is perpendicular to the direction of transmission, and may be either in or perpendicular to the plane of polarisation : in either case it may be repre- sented by a line perpendicular to the plane of polarisation. In the above experiments of Fresnel and Arago, the rays were allowed to interfere during aerial transmission ; it may reasonably be assumed, however, that the same kind of symmetry with respect to two perpendi- cular planes obtains for a plane-polarised ray as transmitted within any crystalline medium : the assumption is not only reasonable on general grounds, but is consistent at once with all known experimental results and with the requirements of the most recent version of the elastic theory (see also pages 17, 32). It is not the only assumption which can be made: Fresnel himself was led by the hypothesis of an incompressible elastic ether to infer that a plane-polarised ray transmitted within a bi-refractive medium is in general symmetrical to only a single plane, perpendicular to the plane of polarisation ; he inferred, in fact, that the vibrations of the ether lie in the transverse plane and are in general oblique, not perpendi- cular, to the direction of the ray. That Fresnel felt the unsatisfactory character of the inference, in the absence of any experimental proof of the obliquity, will be seen on reference to the original memoir. 1 (b) If the two polarised rays which have been obtained from a ray of common light by means of a crystal of calcite are transmitted along the same line, it is found that the resultant effect is again that of a single ray of common light : hence we may infer that in common light, as in plane-polarised light, the vibrations are perpendicular to the direction of transmission of the ray. Representation of the resultant effect of the simultaneous transmission along the same line of two or more plane-polarised rays having different directions of planes of polarisation. 8. ('i.) If the component ra;s have the same wave-length and velocity. (1.) The periodic change at any point of a plane-polarised ray being kine- matically represented by a vibration perpendicular to the plane of polarisation, let two rays be transmitted with the same velocity along the same line, having different directions of the plane of polarisation : and, in the first place, let the algebraical expressions for the corresponding changes be respectively y = rtsinj -~(vt x) + [and z = l sin] ^-- (vl .r) + *. A ^ v X 1 Loc. cit. ; 1827, p. 158. 80 THE TRANSMISSION OF LIGHT IN CRYSTALS. Assuming as before the principle of superposition, the effect of trans- mitting both rays simultaneously will be represented by the motion of a point of which the co-ordinates y and z, measured along the normals of the planes of polarisation, are given by the expressions Eliminating -^ (vt-v), we find Hence the point, of which the position at any instant represents the resultant disturbance at that instant at a corresponding point on the line of transmission, describes in general an ellipse, of which the magnitude and position relative to the planes of polarisation of the original rays are independent both of x and t : all the ellipses are thus equal and parallel, and form a cylinder of which the base is elliptical, and the axis is in the direction of transmission. It will be found that the direction in which the point moves round the ellipse is determined by the relative phases of the two rays. The composite or resultant ray of light due to the co-existence of the original rays is said to be eHiptically polarised ; a ray of which the characters are related to a cylinder with elliptical base must differ from a ray of common light, of which the characters are the same on all its sides. (2.) If the rays have the same intensity, and their difference of phase is measured by the angle between their planes of polarisation, a = b, and a /3 is equal to the angle between the directions of y and ^ : in this case the ellipse becomes a circle, and the cylinder becomes one with a circular base. The composite ray is then said to be circularly polarised. Such a ray is similarly related to every plane passing through it, and yet differs from one of common light : for the motion of the representative point is not symmetrical to a plane, and the characters of the ray may conceivably differ with the direction in which the circle is described by the ideal point. In fact, experimental methods enable us to distinguish, not only between a ray of common light and one which is circularly polarised, but between two circularly polarised rays of which the motion of the ideal point is in opposite directions. (3.) If sin (a /3) = 0, that is to say, if the difference of phase is zero or a multiple of TT, the ellipse becomes one or other of the two straight lines SUPERPOSITION OF PLANE -POLABISED RAYS. 81 1 == 0: hence the resultant ray is itself plane-polarised; the direc- tion of the plane of polarisation depending on the ratio a : />, and thus being determined by the relative intensities of the two component rays. Conversely, such a single plane-polarised ray of simple colour is equivalent in its effects to two such plane-polarised rays of the same simple colour, transmitted along the same line with the same velocity, and with their planes of polarisation in any assigned directions. If the two assigned directions be perpendicular to each other, and 6 be the inclination of one of them to the plane of polarisation of the original ray supposed to bo represented by the expression the two equivalent rays are represented respectively by the expressions y = a sin 6 sin i ~ (vt .r) + a [ ' A \ z = a cos sin J ~ (vt cu) + a } ; \ A, ) for the resultant effect of these two rays is such that ?/ = tan 0, a constant quantity, whatever be the time or the position of the point in the line of trans- mission. (4.) Further, it will be seen that any number of such rays of the same simple colour transmitted along the same line with the same velocity but with different phases, amplitudes and planes of polarisation, will have a resultant effect identical in general with that of a single elliptically polarised ray of the same simple colour, transmitted along the same line with the same velocity. For let the simple rays be severally represented by the expressions ?/, = ^sin | ~ (vt .r) + 12*r/ t \_i (vt .r) + 27T ^ (vt -.<) + , and let the inclinations of the respective planes of polarisation to a fixed 82 THE TRANSMISSION OF LIGHT IN CRYSTALS. plane of reference through the line of transmission be 1} 2 , ..... 6,,. Each single ray being equivalent in effect to two rays ^ith perpendicular planes of polarisation, one of them coincident with the fixed plane of reference, the whole system of rays is equivalent to the following two systems : y=a 1 BiD. OiSiu -(vtaM-ai \+ ...... -\-a n sin O n sin j ~(vt A ! A ^=rt 1 cos0 1 sin - (r .T)+ 04 r+ ...... -\-a n cos O n sin j ~ (vtx) -\-a n t ; J I A J - A all the members of each of these systems having a common direction of plane of polarisation. As each system is equivalent to a single plane-polarised ray (Arts. 3 and 5), the two systems are together equivalent in general to a single elliptically polarised ray. (5.) Whether the resultant ray be elliptically, circularly, or plane- polarised, the resultant change has the same period as the change for each component ray, and is thus of unaltered colour. (6.) At a given instant, the ideal points representing the state at all points of the resultant ray lie on a spiral curve surrounding the elliptical or circular cylinder, if the ray be elliptically or circularly polarised, and on an undulating curve (the curve of sines) in the transverse plane, if the ray be plane-polarised. (b.) If the component rays differ in wave-length or velocity. If the two component rays differ in wave-length or velocity of trans- mission, the resultant effect is still represented by the combined expressions -*)+) y = a sin but it is not periodic at all unless the ratio : is commensurable : and X \ even in that case the curve described by an ideal point is not a conic section. The resultant effect can only be that of a plane-polarised ray if the ratio of y to ~, and therefore of sin ! -~(vt .r)-Ki | to sin \ -^ (vt .r)-f-/3 [ (A ) I A j is independent of the time : but if either v or A is different from v' or A' respectively, this constancy is impossible, whether the planes of polarisation cf the original rays arc real or imaginary. HETEROGENEITY OF EACH RAY. 83 Discrepancy of observed and calculated results. 9. But the above calculation of the resultant effect of the simultaneous transmission along the same line of two plane-polarised rays of the same colour with planes of polarisation at right angles to each other is in direct disagreement with the experimental result recorded in Art. 7/>, for the result of superposition of the two plane-polarised rays obtained from a ray of common light by means of a bi-refractive crystal is not an elliptically polarised ray, but a ray of common light having identical characters on e\ery side. "We are thus led to inquire how far the constancy of character of the periodic changes at points in the same ray has really been established by experi- ment. In fact, the annihilation-effect (p. 10) of two rays of identical charac- ter has only been established for a transference of one of the rays through a distance of at most 50,000 wave-lengths : the wave-length in air for sodium- light being nearly j-^-^ millimetres, the above distance is nearly 80 mil- limetres or about one inch : as light is transmitted through air at the rate of 186,000 miles a second, a distance of one inch corresponds to the lapse of only 11 ?84 9601)00 th part of a second * The discrepancy disappears if a ray is assumed to consist of a scries of independent sets of ivaves of the same length. 10. For the sake of a numerical example, let us imagine that two given rays are absolutely identical in character ; that each ray^consists of a series of sections ; that each section consists of at least a million similar waves, but that the waves of one section are absolutely independent of those of every other, except that they have the same period and are transmitted with the same velocity. Let the constant sections of one ray be A l B lt B-^C^ C 1 D 1 ^i^i> and the identical sections of the other raybe^ 2 B 2 , .B 2 C 2 , C Z D. 2 ia^a consider the resultant effect of transmitting both heterogeneous rays simul- taneously along the same line. (1.) If the initial points^!, A 2 coincide, the vibrations are in unison at every point of every section, notwithstanding the heterogeneity of each ray. (2.) If the ray <4 2 / 2 be moved parallel to itself along its own direction through the distance , the two rays will annihilate each other SB at all points where identical sections are superposed, but will in general fail to do so in the regions \vhere different sections overlap ; that is, for a 84 THE TRANSMISSION OF LIGHT IN CRYSTALS. distance at the end of every section. Hence at any given point there a will be annihilation while at least 999, 999^ waves pass by, and more or less unison while half a wave is passing the same point. (3.) In the same way, if the ray A. 2 Z 2 be moved parallel to itself along its own direction through the distance 50,000^ wave-lengths, the two rays will still annihilate each other at all points where identical sections are superposed, but will in general fail to do so in the regions where different sections overlap ; that is, for a distance 50,000 wave- lengths at the end of each section. Hence, at any given point, there will be complete annihilation while at least 949,999| waves pass by, and more or less unison while 50,000| waves are passing the same point : in other words, instead of complete annihilation, there is more or less light during at most -j^ih part of the time : the light will be apparently continuous, but its intensity will not exceed the -^th part of the maximum joint effect of the two rays. The variability of the periodic character will thus account for the appreciable diminution of the interference- effect when one of the rays is moved parallel to itself through a considerable number of wave-lengths. In the following pages we shall only need to consider sets of waves be- longing to a single section of constant periodic character, and may thus proceed as if the constancy of character were really a property of the whole ray. The same assumption accounts foi" the remarkable fact that rays of the snne simple colour, but obtained from different sources, cannot be made to annihilate each other. 11. Hitherto, for simplicity, we have left unmentioned the remarkable fact that rays of light of the same simple colour, whether common or plane- polarised, cannot be made to annihilate each, other if they have been derived from different sources. This is quite inexplicable if a ray is assumed to have constancy of periodic character throughout its extent ; but it is immediately accounted for by the assumption arrived at in the preceding Article : if a ray consists of a series of independent sets of waves, it is physically impossible for two rays from different sources to be identical in their characters. For a plane-polarised ray, only the amplitudes and phases will differ in the different sets. We have seen that two plane-polarised rays of constant periodic cha- racter throughout would give an elliptically polarised ray of which the ellipses would have a definite magnitude and position dependent on the am- A REPRESENTATIVE FORCE. 85 plitudes and phases of the component rays : if each of the plane-polarised rays, instead of being of constant periodic character throughout, consists of independent sets of waves, the resultant effect will generally be a rapid succession of elliptically polarised sets, the magnitudes and positions of the ellipses changing as different sections of the plane-polarised rays become superposed ; the resultant ray will thus be generally identical in character on all its sides, as far as observation can detect. Not only is the assumption of varialility of periodic character necessary, but a constancy of periodic character could not be jthysically maintained. 12. A simple pendulum, disturbed and then set free to oscillate under the constant action of gravity, soon comes to rest if allowed to communicate its motion to a surrounding medium : to maintain the oscillations, the pendulum requires to be repeatedly disturbed, and each impulse may change the phase and amplitude, and possibly also the direction of the vibration. In the same way, the vibrations of character at the points of a luminous body must be maintained by the repeated action of something analogous to an impulsive force. It is impossible to imagine that the representative impulse can always have the same magnitude and direction, and occur at the particular instant when the vibration is in a particular phase. Hence the vibration must, of almost absolute necessity, be different in its ampli- tude, phase, or direction, after ever} 7 impulse. Further, as already remarked in Art. 6, any luminous source available for experiment is not a geometrical point, but an area of appreciable mag- nitude, and the resultant effect at any point is due to the superposition of the effects of rays transmitted from every point of the luminous area : even if it were possible that the vibrations at a single point could be maintained constant in periodic character, it is inconceivable that the constancy of periodic character could be maintained at points belonging to an appreci- able area. A representative force. 13 . In the case of a plane-polarised ray of constant character throughout the part considered, the vibratory motion of the representative pointy is thus the same for all points P in the line of transmission, and only the phase of the vibration differs at different points at a given instant : hence the expres- 2?r sion y = a sin-^ vt, which represents the change of state at the time t at the point for which -y a = 0, also represents the vibration at any other point of the ray, if we have due regard in every case to the epoch from 86. THE TRANSMISSION OF LIGHT IN CRYSTALS. which the time is measured. The general expression for the law of the change at any point of a plane -polarised ray has been deduced on the as- sumption that the variation of the state with the time is exactly the same as the variation of position of an isochronous pendulum ; or, what is the same thing, of a particle of unit mass vibrating in a straight line and attracted towards an origin in the line by a force of which the magnitude is proportional to the distance therefrom. For the velocity u of the attracted particle at the time t being -~, the accelerativc force at the same instant is -,- or ^ : by hypothesis the force is attractive, and is measured by/ 2 times the distance, or by / 2 //, where/ is a constant &9 quantity: hence ^ = f*y. It is easily seen that y= Bsiu(ft + /3), in which B and ft are both independent of the time, is a solution of this differential equation : for differentiating once we have -^=//>cos(/-}-/3), and, differentiating a second time, C -~ = -f 2 Bsin(fi +/3) = -fy. ttt If the time be measured from an epoch of passage through the origin, the constant ft is zero and the expression becomes y=Bsinft. Hence in the case of plane-polarised light, the vibratory motion of the 27T representative pointy, being expressed by the relation y = a sin vt, is A. identical with that of a particle of unit mass attracted towards the origin by a force which is measured by ry times the distance. A" Even if the actual change of state at the point P were an oscillatory rotation of an ethereal particle about a diameter, as suggested by Rankinc, 1 the above kinematical representation would still hold : in that case, the direction of the line Pp would represent that of the axis of rotation of the ethereal particle at P, and the distance Pp would represent the angular disturbance at the given instant. Or again, the real change may be an electro-magnetic disturbance, what- ever that may be. The representative force is dependent on the luminous sourer. 14. But it will be obvious on reflection that the relation between the 1 Philos. Magazine; 18uo, ser. 4, vol. 6, p. 403. RELATION TO THE LUMINOUS SOURCE. 87 distance of the ideal particle, and the ideal force which acting upon the ideal particle would cause a vibration isochronous with that of the periodic change involved in the transmission of the given ray of light, is generally independent of the specific properties of the transmitting medium. The ratio being 4?rV : X 2 depends only on the ratio \ : v, that is to say, on the period of the vibration or the colour of the light. Now simple light generally retains its colour after transmission through any number of different media ; it is only in fluorescent bodies that the colour of the light or the period of the change suffers alteration : whence we must infer that the period of vibration at any point of a ray, and thus the ratio of the ideal force to the distance, depends in general, not on the specific properties of the medium, but on the period of vibration of the change at the luminous source. The change of colour frequently observed after the passage of light through a medium is really due to the heterogeneity of the colour of the original light, and to the change of relative intensity (not period of vibration, or colour) of the component simple rays. Further analogy mtfi sound. 15. The same is true in the case of sound. Here, again, the trans- mission of a simple note causes a periodic change which may be represented algebraically by the same expression y=aBm \ - (vt &) + a /, and I A. ; kineinatically by the same to-and-fro motion of a particle attracted to an origin with a force measured by times the distance : and the constant A" ratio 47rV 2 : X 2 depends only on the period of the vibration or the note of the sound, and thus on the source of the sound, not on the properties of the transmitting medium. Now the actual change of state at any point of a line of transmission of sound is known to be generally a to-and-fro motion of a particle of the medium^ and the ideal particle and its motion may generally be taken to coincide with the real particle and its motion. But the magnitude of the representative force which acts on the ideal particle must not be confused with that of the elastic force which is evoked at the same point by the disturbance of the sound-transmitting medium : the representative force depends on the period of vibration at the source ; the elastic force evoked by a given displacement depends on the specific properties of the medium : the resultant force acting on the real particle depends, not only on the specific properties of the medium, but on the continued action of the vibrating source. 88 THE TRANSMISSION OF LIGHT IN CKYSTALS. The representative force in the case of the vibration of an clastic ether of which the effective density depends on the direction of the vibration. 16. To take another example : in the latest hypothesis as to the properties of an elastic luminiferous ether, it is assumed that the actual and effective elasticity of both volume and figure and the actual density of the ether are the same for all directions in a biaxal crystal, hut that the effective density varies with the direction of vibration and is related to three mutually perpendicular lines. Hence, if the ether vibrates freely after disturbance parallel to one or other of these lines, the period of vibration will depend on the direction of the disturbance ; for, though the effective elasticity is the same for each direction, the effective density, or effective mass to be put in motion, is different : and the ideal force, which acting on an ideal particle of unit mass gives a synchronous representative vibration, will have a different relation to the distance for the three directions of disturbance, although the ethereal elasticity, both actual and effective, is assumed to be really identical for all directions. A fallacy. 17. We are now in a position to recognise the fallacy of a method which has been used for the derivation of Fresnel's wave-surface from the properties of an incompressible elastic ether. It is first proved that the elastic force evoked by a unit displacement along a line OP, which is the radius vector of a certain ellipsoid, if resolved along the direction of displacemert OP, is . , : that if OP is an axis of a section of the ellipsoid, the other component is perpendicular not merely to OP but also to the plane of the section : that if the section has the direction of the wave- front, the second component is without effect owing to the incompressi- bility of the ether : that the effective clastic force for unit displacement is thus TT. It is then tacitly assumed that the effective elastic force is identical with the above representative force (which is measured by -^7 A" 1 4 7r 2 <; - i times the distance) : hence it is inferred that -ra~ = 7ry,2. It is next A wrongly assumed that the wave-length A isalways the same for rays of th colour transmitted in the same medium, and that A in the above relation is thus a constant : whence it is concluded that the velocity varies inversely as OP. That the proof is fallacious is clear from the last Article, iii which it hus been shown that, in the representative vibration, the relation of the Heal force GENERALLY ONLY TWO RAYS TRANSMISSIBLE IN A GIVEN DIRECTION. 89 to the ideal distance is independent of the specific properties of the medium and depends on the luminous source. Indeed the assumption of the constancy of A. is inconsistent with the conclusion, namely that v differs with the direction of vibration ; it is obvious that the period - and the wave-length X cannot be both constant if v be variable : the colour really depends on the period of tibratwii, rot solely on the wave-length. Frcsuel himself proceeded in a different way, and assumed a relation founded on the analogy of a line of vibrating ethereal particles to a vibrating string. In general^ if a plane-polarised ray in transmissible in a given direction, the plane of polarisation can have at most two different positions. 18- We have seen (Art. 8) that if two plane-polarised rays of the same wave-length can be transmitted along the same line with the same velocity but with different positions of the plane of polarisation, they may be identical in effect with a single plane-polarised ray transmitted along the line with the same velocity but with an intermediate position of the plane of polarisation ; the direction of the latter being determined by the ratio of the amplitudes of vibration of the component rays : conversely, the effect of a single ray of given plane of polarisation and simple colour is identical with that of two rays of the same simple colour transmitted along the same line with the same velocity, and with their planes of polarisation in any assigned positions. Now a plane-polarised ray can be transmitted along the line of inter- section of two planes of physical symmetry of the medium, for the planes of symmetry of the plane-polarised ray and the planes of symmetry of the medium may be taken to coincide : but the velocity of the ray will depend upon the position of the plane of polarisation, if the physical rela- tions of the medium relative to the two planes of symmetry are different. If the latter be the case, as for instance when the line is an axis of sym- metry of an ortho-rhombic crystal, no ray having a plane of polarisation oblique to the symmetral planes of the crystal can be transmitted along it : for such a ray, if transmissible, would be kinematically equivalent to two rays transmitted along the line with the same velocity, each having its plane of polarisation coincident with a different plane of symmetry ; two rays can be actually transmitted with these positions of the plane of polarisation, but that their velocity should be equal is in general phy- sically impossible. In exactly the same way it follows that if along any line, whether an 00 THE TEANSMISSION OF LIGHT IN CRYSTALS. axis of symmetry or not, a plane-polarised ray can be transmitted with its plane of polarisation in two different positions but with different velocities in the two cases, a third position of the plane of polarisation is physically impossible. The refraction of the medium cannot be higher than double. 19. Hence, for a given direction of transmission in such a medium, a plane-polarised ray cannot have more than two different velocities : and the medium cannot present more than double refraction ; for, according to the undulatory theory, whatever the nature of the physical change, the direction of the refracted ray is dependent upon its velocity. Degree of the equation of the ray -surf ace. 20. A diameter of the ray-surface for such a medium will thus inter- sect the surface in at most four real points, two on each side of the origin ; and the equation of the ray-surface cannot be of a degree higher than the fourth, if it be granted that the above method of proof excludes the exis- tence of imaginary velocities and imaginary points of intersection of a real line with the surface. In fact, even if there be two imaginary positions of the plane of polari- sation for a given real direction of ray-transmission, the imaginary velocities must be in general unequal, since the two planes will be differently related to the crystal and will thus correspond to different crystalline properties, whether real or imaginary. But even if the planes of polarisation be imaginary, the difference of the imaginary velocities of the two plane- polarised rays prevents the resultant effect from being that of a single plane-polarised ray with an imaginary plane of polarisation (Art. 8#). The transmissibility of even a single plane-polarised ray is not a physical necessity ; but if one position of a plane of polarisation be possible, there is a second at right angles with the first. 21. We may remark that it is not a physical necessity that a plane- polarised ray should be transmissible at all : a plane-polarised ray cannot be transmitted, for instance, along the morphological axis of a crystal of quartz. As a plane-polarised ray is symmetrical to two planes, the plane of polarisation and the transverse plane, it would seem that if the characters of a crystal admit of one symmetral plane of the ray having a given position, they must admit of the other symmetral plane having the same position : in other words, for a given direction of transmission, if there is TilE VELOCITY- FACTOR. 91 one possible position of the plane of polarisation, there is a second at right angles to the first. The same result is later arrived at in another way and the positions of the perpendicular planes are determined (Art. 40c). Transmission of a my along an axis of tetragonal or hexagonal symmetry, 22. On the other hand, the morphological axis of a tetragonal or hexa- gonal crystal is the intersection of two or more symmetral planes for which the physical relations are identical : hence along such a line it 'is physically possible to transmit two rays having the same velocity and different planes of polarisation, and thus having a resultant effect identical with that of a single plane-polarised ray. The amplitudes of the component rays being arbitrary may be so adjusted that the equivalent single plane- polarised ray has any plane of polarisation whatever : and it follows that, along the morphological axis of a given tetragonalor hexagonal crystal, a ray may be transmitted with any direction of the plane of polarisation, but in each case with the same velocity. The velocity -factor. 23. The velocity of transmission of a plane-polarised ray of given colour is found to depend on the properties of the medium : since the vibration is in only a single direction, we may assume that the velocity of transmission corresponding to a given direction of vibration depends solely on the properties of the medium relative to the direction of the vibration. To avoid confusion of ideas, let the action of the medium, in so far as it affects the velocity of a ray of given direction of vibration, be said to be due to a velocity-factor ; the magnitude of the factor depending on the properties of the medium for the direction of the periodic change or vibration. The velocity -factor is necessarily the same for all directions perpendicular to an axis of tetragonal or hexagonal symmetry. 24. We have shown, from principles of mere symmetry of the medium and superposition of changes, without regard to their physical character, that along the morphological axis of a tetragonal or hexagonal crystal a ray is transmissible with the plane of polarisation in any azimuth whatever, and that the velocity of transmission of the ray is always the same : hence, for all directions of vibration perpendicular to the morphological axis of a uniaxul crystal, the velocity-factor has the same magnitude. It follows that the symmetry of the velocity-factor, at any rate for directions of THE TRANSMISSION OF LIGHT IN CRYSTALS. rectilinear vibration lying in a plane perpendicular to the tetragonal or hexagonal axis of symmetry, is of a higher order than that of the mor- phological development. The corresponding geometrical character is worthy of remark, namely, that in a parallclepipedal system of points every plane of the system passing through an axis of tetragonal or hexagonal symmetry is a plane of symmetry for the planes and lines, though not for the points, of the system : the symmetry of the system relative to such a plane being in genera- " symmetry of aspect," and not absolute. 1 T/'ansmission of a ray in a direction lying in a plane of general symmetry but oblique to an axis of tetragonal or hexagonal symmetry. 25. Consider the case of a ray transmitted in one of the planes of sym- metry S of a tetragonal crystal, but in a direction oblaque to the morphological axis. Either plane of symmetry of the plane-polarised ray may be taken to coincide with the plane of symmetry S of the crystal : this is confirmed by experiment, for these directions of the planes of polarisation of a ray are found to be physically possible. But if the plane of polarisation of the ray is coincident with the plane of symmetry S of the crystal, and the vibration is assumed to be perpendicular to the plane of polarisation, the vibration is perpendicular to the morphological axis, whatever the position of the ray in the plane : hence, according to the preceding Article, the velocity-factor, and therefore the velocity of the ray, will be the same for all ray-directions in this plane, and one curve of intersection of the ray- surface with the plane of symmetry S of the crystal will be a circle. On the other hand, if the plane of polarisation of the ray is normal to the plane of symmetry S of the crystal, the vibration will be in the same plane of s} 7 mmetry S and in a direction oblique to the morphological axis : the physical characters belonging to the direction of the vibration, including the velocity-factor, will thus vary with the direction of the ray, and the velocity- curve corresponding to those rays of which the plane of polarisa- tion is normal to the symmetral plane S of the crystal will not be circular : the curve will be symmetrical, however, both to the morphological axis and a line perpendicular to it, for they are directions with respect to which all the characters of the crystal are symmetrical. Further, the second curve will touch the first at its points of intersection with the morphological axis : for the two directions perpendicular to that line, and lying respectively in and perpendicular to the plane of symmetry S, are by 1 II. J. S. Smith ; Philosophical Magazine, 1877, ser. 5, vol. 4, p. 18. AN ORTHO- RHOMBIC CRYSTAL. 93 hypothesis similar in all their relations, and correspond therefore to the same velocity-factor ; hence both curves meet on the morphological axis, and therefore touch each other, for the morphological axis divides each curve symmetrically. But since the equation of the ray-surface has been shown to be of a degree not higher than the fourth, and the equation of one curve of intersection, a circle, is of the second degree, that of the other curve will likewise be of the second degree, and therefore represent an ellipse for the curve is closed and has unequal diameters. This result agrees with the experimental discovery made by Huygens. Transmission of rays along the axes of symmetry of an ortho-rhombic crystal. 26. Take next the case of an ortho-rhombic crystal. In the first place, as shown in Art. 18, a ray can be transmitted along any of the axes of symmetry, and have its plane of polarisation coincident with either of the symmetral planes of the crystal which intersect therein. The three axes of symmetry being independent of each other in all their physical relations, the velocity-factors will be independent ; and vibrations paral- lel to the several axes will thus in general correspond to different veloci- ties of transmission. Let the velocity corresponding to an axis OX, OY t or OZ, considered as a direction of vibration, be denoted by a, b, or c respectively : then two rays are transmissible along OX with velo- cities b and c, and planes of polarisation normal to OF and OZ respec- tively : two rays are transmissible along OY with velocities c and a, and planes of polarisation normal to OZ and OX respectively : two rays are transmissible along OZ with velocities a and 7>, and planes of polarisation normal to OX and OY respectively. Transmission of rays in a symmetral plane of an ortho-rhombic crystal. 27. Again, as far as directions lying in the plane of symmetry OXZ are concerned, there is no essential difference between an ortho-rhombic and a tetragonal crystal, if OZ is the morphological axis of the latter. The essential difference between two such crystals is that in one of them (the ortho-rhombic) the third axis of symmetry OY is independent of OX in its physical relations, and in the other (the tetragonal) is iden- tical therewith. Hence we may infer that in the symmetral plane OXZ of an ortho-rhombic crystal, a ray is transmissible in any direction with its plane of polarisation either coincident with or perpendicular to that plane. Also, as in the case of a tetragonal crystal, the intersection of the ray- 94 THE TRANSMISSION OF LIGHT IN CRYSTALS, surface with the symmetral plane will be a circle and a concentric ellipse : but, in the ortho-rhombic crystal, the circle and ellipse will be independent of each other in magnitude, since the velocity-factor for the direction of vibration OY is independent of those for the directions of vibration OX, OZ. Intersections of the ray-surface ivith the symmetral planes of an ortho-rhombic crystal. 28. The intersections of the ray-surface with the axial planes OYZ, OZX, OXY of an ortho-rhombic crystal will thus be given by the following equations : (f + * 2 - a a ) (by + cV - &V) = 0, (,2 +a . 2 _^ ( c v+ aV-cVr 2 ) = 0, (a* _|_ y a _ c 2 ) ( V -f by - aW) = 0. General equation of the ray -surf ace for an or t ho -rhombic crystal. 29. The equation of the ray-surface itself must be of the form (jf+s* _ 2 ) (by +<*,? - W) +.1- 0(.^c) = 0, since it reduces to the first expression when x is made zero. But ac- cording to Art. 20 the quantity x (-vyz) cannot consist of terms of degrees higher than the fourth : further, the surface being symmetrical to the axial planes, its equation can only involve even powers of x, y, z : hence the only terms which can enter the expression .r 0(.v/?) are .r 4 , s 2 ,i >2 , ay and .r 2 . The general equation is thus of the form or, multiplying out, Ax* + iy + c V -f (/. 2 + c- 2 ) / : 2 + BzW + 0*y + Da? - 1>* (<* Also, it is evident from the equations of the curves of intersection with the three axial planes that a?, y, z, and a t />, c, are simultaneously cyclically interchangeable (Art. 28); hence A=a 2 ; J5 = c 2 + rt 2 ; (7=a 2 + i 2 ; D = -a*(b* + c*). Substituting these values, the equation becomes ^ 2) (aw+tfyi+c^)^ (* 2 +c 2 ) a;2 62 (c*+a?) y*-c* (a?+l*) z*+aWc* = or, multiplying by r 2 , or rt,V (/ 2 - V s ) (r 2 - c 2 or cf r- r e wnich is Fresnel's equation of the ray-surface. A MONO -SYMMETRIC OR ANORTHIC CRYSTAL. 95 Ike ray -surf ace for a mono-symmetric or anortliic crystal. 30. (a) Next consider the case of a crystal which admits of the trans- mission of a ray of plane-polarised light in any direction, but presents only a single plane of geometrical and physical symmetry, and thus belongs to the mono-symmetric system : let the normal of the plane of symmetry be OF. Since, from a purely geometrical point of view, a mono- symmetric crystal may be regarded as a homographic transformation of an ortho- rhombic crystal, it first suggests itself that the ray-surface for a mono- symmetric crystal may be such as would result from a corresponding trans- formation of the ray-surface for an ortho-rhombic crystal. That the analogy is imperfect, however, is evident from the fact that there is no corresponding distortion of the planes of polarisation ; whatever the direction of ray- transmission within the mono-symmetric crystal, the planes of polarisa- tion of the two transmissible rays are perpendicular to each other (Art. 21). 1. As in Art. 27, any ray whatever lying in the plane of symmetry can have that plane for either its plane of polarisation or its transverse plane : hence, exactly in the same way as before, it follows that the plane of symmetry intersects the ray-surface in tsvo curves, the one a circle, the other a con- centric ellipse : the former corresponding to the rays which have the syrnmetral plane for the plane of polarisation, the latter to the rays for which the plane of symmetry is the transverse plane. If OX, OZ, be the axes of the ellipse, a ray transmitted along OX will thus have its representative vibrations parallel to either OF or OZ ; and a ray transmitted along OZ will have its vibrations parallel to either OF or OX. 2. A ray transmissible along the line OF can have its plane of polari- sation in only one or other of two directions of which the normals are perpendicular both to each other and to the line OF (Art. 21). 3. Since the elliptic and circular sections of the ray- surf ace made by the plane XOZ are both of them symmetrical to the lines OX, OZ, while the plane XOZ is a plane of general physical symmetry of the crystal, and its normal OF is an axis of general symmetry of diagonal type, we may reasonably assume that for this particular property (so long as there is no variation of colour or temperature) the planes FO^T, YOZ, are themselves planes of symmetry of the crystal ; in which case, the lines OX, OZ, will be the directions of vibration of the two rays transmissible along the axis OF. 4. For the given colour and temperature, the circumstances are identical, for this particular property, with those of an ortho-rhombic crystal having OX, OF, OZ, for axes of symmetry : and the ray- surface 96 THE TRANSMISSION OF LIGHT IN CRYSTALS. will thus for a mono -symmetric crystal have the same general form as for an ortho -rhombic one. (b) The general form of the ray-surface, being quite unaffected by the degradation of the symmetry from the ortho-rhombic to the mono-sym- metric type, is clearly independent of the type of symmetry altogether : the general form will therefore be the same even for an anorthic crystal. The difference in the type of symmetry tLus affects, not the general form of the ray-surface, but only the constancy of the directions and rela- tive lengths of the axes of the surface for different colours and tempera- tures. An axis of general symmetry of the crystal is necessarily an axis of symmetry of the ray-surface whatever the colour of the light or the temperature of the crystal (p. 21). The form of the ray -surf ace is independent of the physical character of the periodic change. 31. The rigorous accuracy of the form assigned to the ray-surface by Fresnel is thus a necessary consequence of the general features of perpendi- cularly transverse vibrations, independently of the physical character of the change. And although, as in the case of an incompressible elastic ether with effective rigidity dependent upon the direction of vibration, the same form of ray- surface may result notwithstanding the obliquity of the transverse vibration, this is not generally true. The form of the ray-surface which follows, for example, from a version of the elastic theory of double refraction suggested by Rankine and further developed by Lord Eayleigh is different from that of Fresnel, and only gives the latter as a first approxi- mation. That version, according to which the ether is incompressible and has an effective density dependent on the direction of vibration, involves the general obliquity of the latter to the direction of transmission. 1 In fact, whatever the degree of symmetry of the characters of a plane- polarised ray as transmitted within a medium, the above form of ra}^- surface will result from any hypothesis which has for necessary conse- quence that if one plane-polarised ray is transmissible in a given direction, a second plane-polarised ray is transmissible in the same direction with a different velocity and has its plane of polarisation perpendicular to that of the first. It may be remarked that, in the above reasoning, no assumption as to the molecular constitution of the ether has been necessary. 1 Philosophical Magazine; 1851, ser. 4, Vol. 1, p. 441: 1888, ser. 5, vol. 26, pp. 525, -527. FREE AND FORCED VIBRATIONS. 97 Resilience. 32- It was explained in Art. 14 that the period of the change at any point of a ray of light depends in general on the period of the change at the luminous source, and not on the specific properties of the medium ; but the latter may conceivably affect some or all of the remaining characters of the ray, namely, amplitude and direction of vibration, velocity and direc- tion of transmission through the medium. The property by virtue of which a periodic change of any kind is transmissible through a medium may be denoted by the general term resilience : we may imagine that a disturbance at any point of the medium evokes an opposing resilience of which the magnitude increases with the amount of the disturbance. Optical resilience, in so far as it affects only the velocity of transmission of a periodic change having a given direction, is identical with the velocity-factor for that direction, mentioned in Art. 23. When the periodic change is a vibratory motion such as follows the removal of a compressing or distorting force, resilience is identical with elasticity of volume or figure. From this point of view, the periodic change at any point of a ray of plane-polarised light may be treated as a resultant effect of two forces ; the one an initiatory linear force periodic in its variations, and having a period identical with that of the luminous source ; the other a secondary force or a resilience, evoked by the disturbance produced by the initiatory force. Free and forced vibrations. 33. The periodic change at a point of a ray of light is a forced vibra* tion, resulting from the continued action of the luminous source : it differs from &free vibration, such as would be produced by resilience alone if the luminous source were removed while the medium is in a state of disturbance. A simple case of free vibration. 34. In the simplest possible case of free vibration of a character of a medium, we may imagine that the disturbance at the point is of such a kind that at any instant it can be represented by the length and direc- tion of a straight line y drawn from an ideal particle of unit mass to the point, and that the resilience of the medium can be represented by an ideal force acting in the line of disturbance, tending to diminish the dis- turbance, and proportional in magnitude to the disturbance itself, the proportion being independent of direction : such a medium may be said to be isotropically resilient for the given character. THE TRANSMISSION OF LIGHT IN CRYSTALS. As in Art. 13 we may write wlurc/ is independent of the time and depends on the properties of the medium. A solution of this equation is y = Z>sin(/Y+/3), where B and / } arc constants : the expression represents a vibration of which the period is 2- -TT, since any value of y recurs when t is increased by an integral multiple of that quantity. As already pointed out, such a mode of representation is still possible, even when the actual change is an oscillatory rotation of an ethereal particle (Art. 13). A simple case of forced vibration. 35. But suppose that in tha above medium the vibration at the point is not free bat forced, and that the initiatory force is a periodic one related to tli3 time in the same way as the disturbance at a point of a plane-polarised ray of simple colour ; the initiatory force can in such case be represented by an expression of the form S&inst, where Sand s are constants, and the latter depends only on the period of vibration of the luminous source. As be- fore, the ideal resultant force acting on the ideal particle of unit mass is &# TTJ, and is due to tha superposition of the initiatory force Ss'mst and the resilience -fy : hence gjj = Ssiust -fy. It is easily seen that y^Bs'mst is a solution of this differential equation : for differentiating, we get first -~ KsGOSst, and next -7^ = Z?s' 3 sin*: substituting in the above equation, and dividing by sins, we get o -73 2 ~ / * merely the 'amplitude, not the period or general character of the vibration at the point. The resilience, being / 2 y, has likewise the same period as the initiatory force. B = - 2 : an( * tnus y ~ /> o simtf. Hence the resilience affects " " Transmission of a simple forced vibration in an isoiropicdly resilient medium. 36. If a luminous source is in a state of periodic vibration represented AN /EOLOTROPICALLY BESILIENT MEDIUM. 99 kinomatically by the linear motion of a particle attracted to an origin by a force proportional to the distance, and is surrounded by a medium such that the resilience is represented by a force acting in the line of disturb- ance and proportional to it in magnitude, the changes transmitted through the medium along a given direction perpendicular to that of the vibra- tion may thus be expected to be always in the same plane and have the same period ; no resilient force oblique to the plane containing the direction of ray-transmission and the direction of vibration of the luminous source is evoked by the disturbance : in any direction in an isotropically resilient medium, a plane-polarised ray, if transmissible at all, may thus be transmitted with any azimuth of plane of polarisation whatever. A more general case of free vibration of an ceolotropically resilient medium. 37. As a more general case, we may imagine that in a crystalline medium there are three directions, not co-planar, inclined obliquely or perpendicularly to each other, for each of which a disturbance evokes a resilience which in its effects is represented by an ideal force, contrary and proportional to the disturbance, acting on an ideal particle of unit mass ; the relation of the ideal representative force to the distance of an ideal attracted particle of unit mass being, however, like most other physical characters, different for the three directions : the latter may be termed axes of optical resilience. That such a representation is possible, even in an elastic ether of which the elasticity is the same in all directions, has already been pointed out in Art. 16 : for if the effective density depends on the direction of vibration, the period of a free vibration will also vary with the direction, since although the real accelerative force has the same constant relation to the distance it will have a different effective mass to keep in motion. When it is desirable to emphasise the fact that the resilient force under consideration is the ideal force which would produce an analogous to-and- fro motion of a particle of unit mass and not the statical force necessary to the maintenance of a given state of disturbance, we may conveniently dis- tinguish it as vibrational resilience. If the three constants of vibrational resilience be respectively e\ / 2 , (f, and x, y, z, be the distances which re- present the disturbances parallel to the respective axes at any time t, we have for a free vibration due to a disturbance along each of the axes d\v tfy d*z w=-**'> &=-*&=-?* whence, in the same way as before, + a); y = Bsm(ft+ (3) ; 2 = <7sin(^+ y). 100 THE TRANSMISSION OF LIGHT IN CRYSTALS. According to the principle of superposition of changes, if the direction of the initial disturbance at the point is inclined to the three axes of resilience, the initial disturbance may be resolved along those directions, and the resultant free vibration is such as would result from the com- position of the free vibrations corresponding to the several axial direc- tions. Hence, if tho vibration is freo, the disturbance at a given instant is determined by the above triad of equations. Since the ratios .r : y : z depend on the time, the motion of the repre- sentative particle is not in a straight line passing through the origin. The particle, in fact, describes a curve in three dimensions, and never passes twice through the same position unless the ratios e : f : y are com- mensurable. The quantities e z ,f' 2 ,y~, may be conveniently termed coefficients of optical vibrational resilience : and the medium may be said to be ccolotropically resilient. The coefficients of vibrational resilience are independent of .< and t for the same ray, but even with the same medium may conceivably be different for different rays, and thus vary with the period of the change, or in other words, with the colour of the light. In Art. 42 it is pointed out that obliquity of mutual inclination of the axes of optical resilierce is not met with even in mono-symmetric or anorthic crystals. A more general case of forced vibration of an ccolotropically resilient medium. 38. Consider next a forced vibration of a crystalline medium having three dissimilar oblique or rectangular axes of vibrational resilience as before : assume that the initiatory force at any point of a ray may again be represented by an expression of the form Ssiust, where s is a constant depending on the period of the change at the luminous source. If OP be any line passing through an origin O, and UL, (>J/, (>X, lengths measured along the axes of resilience, be edges of a parallelepiped of which OP is a diagonal, whatever the length OP we have OL=\-OP OM=pOP OX=vOP, where X p. r are constants for a given direction of OP. Fiom the principle of superposition, it follows that the initiatory force tfsinstf acting in the line OP can be resolved into three initiatory forces XSsinsf, p,Ssmst t vSsmst, acting along the axes of x y ~ respectively. In exactly the same way as before we have the following expressions for the several vibrations parallel to the respective axes : \S uS r.S' x = -7, o sins/ ; // = -.:/ ., sins/ ; ; = ., ., sins/ : AN ^EOLOTROPICALLY RESILIENT MEDIUM. 101 where the quantities X p v c' 2 f 2 (f S and s are all independent of the time. Hence the ratios x : y : : are also independent of the time, and the representative particle vibrates in a straight line through the origin. The period of the resultant vibration is identical with that of the initiatory force, but the direction of the vibration is different. If X' p' v' determine the direction of the resultant vibration, x , . / , ^ f* y Further, the axial components of the representative resilient force being fix. / 2 //, n z z, or / J * c/ , . , , XI^SilMrf, r a ~ a sins*, -3 - a sins*, L t> J d (J - S the resultant resilient force will have a direction determined by the ratios Hence the resultant resilient force always acts in the same direction throughout the vibration, but it is inclined to the direction X/u v of the initiatory force and also to the line of vibration X'// l/ ' both of which pass through the origin : further, the resultant resilient force has the same period as the initiatory force. Transmission of a simple forced vibration In an aolotropically resilient medium. 39. In such a medium, therefore, an initiatory linear periodic force having a direction inclined to an axis of resilience and acting at a given point gives rise at that point to a linear periodic vibration in a direction inclined to the initiatory force, and to a resilience of which the resultant effect is represented by a periodic force acting on the ideal particle in a third and constant direction not passing through the given point. Since the periodic change is transmitted through the medium by virtue of the resilience, and action is always equal and contrary to reaction, we should thus expect that, along any line of transmission, the direction of the periodic change will in general vary from point to point of the ray ; and that the transmitted periodic change can only be in a direction lying always in the same plane, if the plane containing the initiatory force and the direction of transmission likewise contains the direction of the resilient force, and therefore also the direction of representative vibration. Consider, for example, the case of a ray transmissible along an axis OX of an ortho-rhombic crystal : from the symmetry it follows that a plane- 102 THE TRANSMISSION OF LIGHT IN CRYSTALS. polarised ray transmitted along OX must have its vibrations parallel to one or other of the dissimilar axes OF, OZ, and that for a ray of given simple colour the velocity of transmission will depend on the direction of vibration. If, however, the initiatory force at the initial point of the ray, though perpendicular to OX, is oblique to the axes OF, OZ, it may be re- solved into two forces, parallel to OF, OZ, respectively, and each maybe regarded as originating a simple plane-polarised ray : the motion of the representative point will be the resultant of the motions belonging to each ray, and will thus be continually changing its direction as the disturbance is transmitted along OX. Case of an ortho-rhombic crystal. 40. (a.) Direction of the resultant vibrational resilience for a given disturbance. For simplicity, let the crystalline medium present three mutually perpendicular but dissimilar symmetral planes, and thus belong to the ortho-rhombic system : the axes of resilience necessarily coincide with the crystallographic axes, the lines of intersection of the symmetral planes. Let X, Y, JZ, be the components, parallel to the axes of co-ordinates, of the representative resilient force corresponding to a disturbance defined by the co-ordinates x' y 1 z' : then A=-*v ; F=-yy ; z=-pv. The direction-cosines of the resultant resilience F are in the ratios X: F: Z-. or V : /y : f /z' . But if an ellipsoid aV+Ty+^^l be of such dimensions that it passes through the point P (x'y'z'}, the direction-cosines of PG the normal of the ellipsoid at the point x'y'z' are likewise in the ratios *V : /y : 0V (Fig. 19). Hence the resultant resilient force due to a disturbance OP acts in the direction PG of the normal of the ellipsoid e V+/y + r/V = l at the point P (x'y'z 1 ) lying on its surface. (b.) Direction oj transmission of a plane-polarised ray of which the direc- tion of the plane of polarisation is given. If OJbe the centre of the ellipsoid and OP the representative direction of vibration, the initiatory force must also lie in the plane OPG which con- tains the direction of vibration and the secondary force : further, the direction of transmission must lie in the same plane (Art. 39 ). The vibration being always perpendicularly transversal to the direction of transmission, the ray Or corresponding to the vibration OP is thus in the plane OPG and perpendicular to OP (Fig. 19). (V.) The planes of polarisation for a given direction of ray are perpen- AN ORTHO-RHOMBIC CRYSTAL. 103 dicular to each oilier, and have directions which can le defined by means of an ellipsoid. It has already been proved (Chapter IV, Art. 24) that if OP be a central radius vector of an ellipsoid, and PG the normal of the ellipsoid at P, the line OP is an axis of the section of the ellipsoid by a plane through OP perpendicular to the plane OPG ; for a given direction of ray Or there are thus two possible directions of vibration OP lt OP 2 , which can be transmitted without change of plane, and they are the axes of the section of the ellipsoid by a plane to which the ray Or is normal. Hence the planes of polarisation corresponding to a given direction of ray are perpendicular to each other and are determined by the above geometrical construction. (d.) Magnitudes of the total and effective vibrational resilience for a given disturbance. If F be the resultant resilience, F*=>X*+7*+Z*=fa;'*+f i y'*-i-$ t z'*. But if OM (Fig. 19) be the central normal to the tangent plane at P (aty'*'), .?V 2 . (Chap. IV, Art. 4). Hence the resultant resilience F is measured by . The resilience being in the direction PG, and the actual vibration in the direction PO, the effective resilience is F cos OPG _ OP ~~ OP' This corresponds to a disturbance of magnitude OP : hence the effective resilience for a unit disturbance in the direction OP is TT- (e.) Pielation between the effective vibrational resilience and the velocity of transmission. In the development of his theory of Double Refraction, Fresnel was compelled to make an assumption as to the relation between the effective elastic force and the velocity of normal-propagation of the corresponding wave, and supported his assumption by reference to the analogy of a vibrating string, In the preceding Articles, all the forces are purely representative, and the assumptions and reasoning founded thereon are really independent of the physical character of the change. But it is clear that the velocity of transmission must depend on the physical character of the periodic change, and that it is impossible to proceed farther and deduce the absolute velocity 101 THE TRANSMISSION OF LIGHT IN CRYSTALS. of transmission without some assumption involving the nature of the change and the constitution of the ether. All that we have been able to suggest hitherto is that the velocity is in some way dependent on the characters of the medium relative to the direction of the vibration, and these characters have been collectively expressed by the term velocity-factor : in other words, it was suggested that the velocities of the two rays trans- missible in the direction Or are determined by some function of the directions of vibration OP ly OP 2 , and thus by some function of the lengths OP lt 0P 2 , for the length of a radius vector of an ellipsoid is determined by the direction. But we have also shown (Art. 29) that, without any assumption as to the real nature of the change, it is possible to determine the velocities r lt r a , of the two rays transmissible in a given direction, in terms of a, b, c, the velocities corresponding to vibrations in the directions of the principal axes : hence the velocity r of transmission along Or is necessarily so related to OP, an axis of the section of the ellipsoid V=l by the plane perpendicular to Or, that the equation a *3 JAjjSt fg ax oy cz 2 ~ represents the ray-surface. There is only one relation between r and OP which leads to this form of ray-surface, namely r=OP: we are thus compelled to infer that the velocity of transmission of a ray is directly pro- portional to that radius vector OP of the above ellipsoid e?J?+J*y*+g*z* = 'l , which has the same direction as the vibrations of the ray : from Art. 40^ it follows that the same relation is expressed by the statement that the ray- velocity is inversely proportional to the square root of the effective resilience for unit disturbance in the direction of vibration. It follows from the above relation that : If a line Or is perpendicular to a central section of the ellipsoid 02^+/y+/ 2 ==l, an( j Qp it Qp 2) are the axes of the section, a plane- polarised ray can be transmitted along Or, having OP l or <9P 2 for the normal of its plane of polarisation and a velocity of transmission measured by OP l or 0P 2 , respectively. That this relation is consistent with the form of the ray-surface arrived at in Art. 29 may be proved as follows : (/.) transformation of the above construction. From draw OM perpendicular to the plane which touches the ellipsoid -2\ Each of these fractions is equal to -, 4 , 2 S fi >a 4 ,. ; that is OR'OM, ^/(e x -\-j y -\-g z ) or unity. FIG. 19. We thus have - = ex'; - = f it follows that l + !>+ = 1 . e 2 y 2 ^2 This is the equation of the locus of the points E, and represents an ellipsoid with the same symmetral planes as ^ 2 +/V 2 +i/ 2;s2=: 1 semi- axes efg instead of --- e f (i 3 (2.) The equation of the plane which touches the ellipsoid ^+ + = 1 at the point E ( 17 ) is 106 THE TRANSMISSION OF LIGHT IN CRYSTALS. If Om be the perpendicular on this plane from the origin, the direction- f. f /-~> ^10 a " o c A MONO- SYMMETRIC OR ANORTHIC CRYSTAL. 107 gA ?/ 2 >2 ellipsoid + -f = 1 is identical with the optical indicatrix v /" for ^n- is the square root of the effective elastic force due to a unit displacement in the direction of vibration. It will be found, however, that such a motion cannot actually take place : the particle will only vibrate rectilinearly if the path passes through the origin. If the particle is not moving in a line through the origin, the evoked elastic force will be constantly changing direction ; for at any instant it acts parallel to the normals of the ellipsoid aV 2 + & 2 .'/ 2 + f 2 ~ 2 = 1 at the points where a line joining the particle to the centre meets the surface. Further, the initiatory periodic constraining force is zero, not when the particle is at its position of maximum displacement, but when it is in its position of no disturbance. ELLIPTICALLY OR CIRCULARLY POLARISED RAYS. 109 The transmission of elliptically or circularly polarised rays. 44. Wo have seen above (Art. 8#) that the simultaneous transmission of two plane-polarised rays of the same simple colour along the same lino with the same velocity, but with different directions of planes of polarisa- tion, has for general result an elliptically polarised ray of the same simple colour transmitted with the same velocity : farther, the right-hand or left-hand character of the motion of the representative point round the ellipse depends only on the relation of the phases of the component rays. Hence, in general, an elliptically polarised ray, or, its spscial case, a circu- larly polarised ray, can be transmitted in any direction within a cubic crystal, or along the morphological axis of a tetragonal or hexagonal crys- tal ; and its velocity is independent of its right-hand or left-hand cha- racter. If the velocity of transmission of a plane- polarised ray along a given direction within a crystal is dependent on the azimuth of the plane of polarisation, we have seen (Art. 8#) that an elliptically or circularly polarised ray cannot result from the composition of two plane-polarised rays transmitted along that direction. The transmission of a circularly polarised ray, however, may be possible even when that of a single plane-polarised ray is not so : for instance, a right-hand or a left-hand circularly polarised ray, but not a plane-polarised ray, can be transmitted along the morphological axis of a crystal of quartz. In such case, the velocities of transmission of a right-hand and a left-hand circularly polarised ray of the same simple colour are neces - sarily different : for it will be found on calculation that a right-hand and a left-hand circularly polarised ray transmitted with the same velocity, if superposed, are kinematically identical with a plane-polarised ray, the azimuth of the plane of polarisation of which depends solely on the relative phases of the component rays ; but, according to hypothesis, a plane- polarised ray is incapable of transmission. Such a line will be an axis of optical symmetry, but cannot lie in a plane of general symmetry ; for symmetry to the plane would require a right-hand and a left-hand ray to be transmissible with the same velocity. In fact, if a right and left circular motion of the same radius and period are simultaneously impressed on the same particle, the resultant motion is a vibration along that diameter of the circle to which the two circular motions are symmetrical, namely, the diameter passing through the two positions of the particle which are identical for the component motions. If the two circular motions are transmitted through the medium with the same velocity, their relative phases, and thus the direc- 110 THE TRANSMISSION OF LIGHT IN CRYSTALS. tion of the line of resultant vibration, will be the same at all points of the resultant ray : if they are transmitted with unequal velocities, the line of resultant vibration will have different azimuths for different points of the ray, and the change of azimuth will be proportional to the distance between the given points. Hence it follows that if a plane-polarised ray be incident normally on a plate cut perpendicularly to the morphological axis of a crystal of quartz, the ray will not be in a state of plane-polarisa- tion within the plate, though it will be so after emergence : the planes of polarisation of the incident and emergent rays will be inclined to each other at an angle which is proportional to the thickness of the plate. SUMMARY. 1. Fresnel's hypothesis that light consists in the vibratory motion of an incompressible elastic ether being untenable, should be abandoned as an educational instrument. 2. The later hypothesis that light consists in the vibratory motion of a compressible elastic ether, of which the elasticity (of volume and figure) is the same for all bodies and for all directions in the same body, and of which the effective density in bi-refractive media is dependent on the direc- tion of the vibratory motion satisfactorily accounts for most of the known optical laws : hence such terms as " axes of optical elasticity," which relate to variation of elasticity, must be discontinued. 3. Even this more satisfactory hypothesis may only be an approximate mechanical analogy, and may eventually be found to be inconsistent with experiment in some of its optical results ; hence it cannot be satisfactorily used as the basis of a correlation of optical characters for the student of crystals ; in fact, though it appears to be fully established that electro- magnetic waves and light-waves differ only in length, an electro-magnetic disturbance seems to be inexplicable as mere vibratory motion of an elastic body. 4. On the other hand, the accuracy of Huygens's construction is now so far confirmed by experiment that it doubtless expresses a Law of Nature. 5. This being the case, it is easily seen that the velocity and polarisa- tion of each of the two rays transmissible in a given direction in a uniaxal crystal can be simply expressed by means of the spheroid alone \ If E be a point on the spheroid, the centre, EN the normal, XOr a line intersecting the normal perpendicularly, the point R corresponds to a SUMMARY. Ill ray transmissible in the direction NOr, with a velocity represented by nv , and having its plane of polarisation perpendicular to RN. liN 6. Generalisation suggests that, in the case of crystals belonging to a lower type of general symmetry, there is a similar correspondence between each ray and a point on an ellipsoid. 7. Experiment confirms the rigorous accumcy of the generalisation. 8. The surface of reference, whether a sphere, spheroid or ellipsoid, may be conveniently denoted by the term optical indicatrix. 9. All the optical characters can be directly deduced from the indicatrix itself, and reference to its polar reciprocal is for this purpose unnecessary : further, it is possible to develop the characters from the consideration of rays alone. 10. The front of a pencil of rays which have started simultaneously from a point is part of the ray-surface ; in the limit, if the pencil is of small aperture and includes a given ray, the pencil-front is part of the plane which touches the ray-surface where the ray meets it : hence, the pencil- front corresponding to the given ray may be briefly designated as the rcuj- front. 11. A plane passing through a ray and perpendicular to its plane of polarisation may be conveniently termed its transverse plane. 12. In such case, it follows that the normal to the ray-front corres- ponding to the ray Or lies in the transverse plane EN Or and is perpen- dicular to Oli, while the velocity of normal-propagation of the front is measured by ^TR' 13. The normal RN is the direction of vibration of the ray corre- sponding to the point R, if the most recent hypothesis as to the properties of an elastic luminiferous ether is true. 14. The so-called primary and secondary optic axes are not axes of symmetry, nor even constant lines, of the crystal : they may with pre- cision be denoted respectively as the optic It-normals and U-radials ; for they are directions in which the two normals drawn from the centre to tangent planes of the ray-surface having the same direction, or the two radii vectores of the ray-surface having the same direction, are respec- tively coincident with each other. A crystal may still be loosely termed bii.ml, when it is merely desired to suggest that the interference-rings shown by a plate in convergent polarised light are rudely like those 112 SUMMARY. which might be expected to be seen if the crystal had two axes, each iden- tical in character with the optic axis of a tetragonal or hexagonal crystal. 15. By help of simple assumptions, which naturally present them- selves and are consistent with all known experimental results, Fresnel's equation of the ray-surface may be deduced from the general principles of undulations, without regard to the physical character of the periodic change. THE END. Printed by Williams and Strahan, 7 Lawrence Lane, Ckeapside, LOAN DEPT 240ct'57CS " imm ed iate recall. rtcx* TTi '65-2 versity of California Berkeley