ELEMENTS OF THE ELECTROMAGNETIC THEORY OF LIGHT ELEMENTS OF THE ELECTROMAGNETIC THEORY OF LIGHT BY LUDWIK SILBERSTEIN, PH.D. LECTURER IN NATURAL PHILOSOPHY AT THE UNIVERSITY OF ROME LONGMANS, GREEN AND CO. 39 PATERNOSTER ROW, LONDON FOURTH AVENUE & 30TH STREET, NEW YORK BOMBAY, CALCUTTA, AND MADRAS 1918 PREFACE. THIS little volume, whose object is to present the essentials of the electromagnetic theory of light, was rewritten, at the instance of Messrs. Adam Hilger, Limited, from my Polish treatise on Electricity and Magnetism (3 vols., Warsaw, 1908-1913, published by the kind help of the Mianowski Institution) . It con- sists principally of an English version of chapter viii., vol. ii., of that work with some slight omissions and modifications. In order to make the subject accessible to a larger circle of readers Section 3 was added. The language adopted is mainly vectorial. This is the chief reason of the compactness of the book which, it is hoped, notwithstanding its small number of pages, will be found to contain an easy and complete presentation of the fundamental part of Maxwell's theory of light. I gladly take the opportunity of expressing my best thanks to Messrs. Hilger for enabling me to submit a portion of my treatise to the English reader. L. S. LONDON, May, 1918. 39R12K CONTENTS. PAGE 1. THE ORIGIN OF THE ELECTROMAGNETIC THEORY ... 1 2. ADVANTAGES OF THE ELECTROMAGNETIC OVER THE ELASTIC THEORY OF LIGHT 5 3. MAXWELL'S EQUATIONS. PLANE WAVES 16 4. REFLECTION AND REFRACTION AT THE BOUNDARY OF ISOTROPIC MEDIA ; E IN PLANE OF INCIDENCE ..... 22 5. REFLECTION AND REFRACTION ; E _L PLANE OF INCIDENCE. NOTE ON THE TRANSITION LAYER 28 6. TOTAL REFLECTION 31 7. OPTICS OF CRYSTALLINE MEDIA : GENERAL FORMULAE AND THEOREMS ......... 35 8. THE PROPERTIES OF THE ELECTRICAL AXES OF A CRYSTAL . 41 9. OPTICAL AXES 43 10. UNIAXIAL CRYSTALS ......... 45 INDEX ...... 47 vii 1. The Origin of the Electromagnetic Theory. The electromagnetic theory of light, now for many years in universal acceptance, was proposed and developed by James Clerk Maxwell about the year 1865.* By elimination, from his classical differential equations, of the electric current Maxwell has obtained, for the "vector potential" 51, t a differential equation of the second order which in the case of a non-conducting isotropic medium has assumed the form tf^=V 2 ?l . - [M] where v 2 is the Laplacian (Maxwell's - v 2 > borrowed from Hamilton's calculus of quaternions). Maxwell's coefficients, the " specific inductive capacity " K, and the magnetic " per- meability" //,, are not pure numbers. Let c be the ratio of the electromagnetic unit of electric charge to the electrostatic unit of charge. Then Maxwell's coefficients are such that, for air (or vacuum), K = 1, fjL = , in the electrostatic system, K = -, fjt. = 1, in the electromagnetic system, c * Phil. Trans., 1865, p. 459 et seq. t reprinted in Scientific Papers. See also Treatise on Electricity and Magnetism, vol. ii., chap. xx. t Which, in absence of a purely electrostatic potential, gives the electric force by its negative time derivative, i.e. in the notation to be adopted throughout this volume, E = - 1 2 THK O1UGIK OF THE Thus, in either sysierij, Kfi -= 1/c 2 , for air. Now, from his above equation which in the case of plane waves, for instance, reduces to Maxwell concluded at once that the velocity of propagatio of electromagnetic disturbances should be in any medium, and therefore, in air, v = c. Thus Maxwell has arrived at the capital conclusion that '" the velocity of propagation in air [or in vacuo] is numerically equal to the number of electrostatic units contained in an electromagnetic unit of electric charge ". .The dimensions of this " number " c, or ratio of units, are obviously those of a velocity. For, by what has just been said, we have the dimensional equation [c 2 * 2 ] = [a; 2 ] where # is a length and t a time. Now, the experimental measurements of Kohlrausch and Weber,* famous in those times, have given for the ratio of the two units of charge the value c = 310740 km. sec." 1 = 3-107 . 10 10 cm. sec." 1 , or rather, after account has been taken of W. Voigt's correc- tions (Ann. d. Phys., vol. ii.), 3'Hl . 10 10 cm. sec." 1 . Max- well quotes also the value obtained from a comparison of the units of electromotive force t by William Thomson (1860), * Kohlrausch and Weber, Elcktrodyn. Maassbestimmungen, etc ; W. Weber, Elektrodyn. Maassbestimmungen, insbesondere Widerstand- messungen. f An electromagnetic unit of electromotive force contains 1/c electro- static units. ELECTEOMAGNETIG THEORY OF LIGHT 3 c = 2-82 . 10 10 cm. sec.' 1 , and the value which he has later (1868) obtained himself from a comparison of the same units, c = 2-88 . 10 10 cm. sec." 1 . These figures Maxwell has compared with those obtained for the velocity of light in air and in interstellar space : 3-14 . 10 10 cm. sec." 1 . (Fizeau). 3'08 ,, . (astronom. observations). 2-98 . (Foucault). The agreement of the light velocity with that ratio of units c has thus turned out to be satisfactory. And, although Maxwell himself states it very cautiously by saying only that his theory is not contradicted by these results, there can be but little doubt that the said agreement has had a decisive influence upon the birth of the electromagnetic theory of light. And later measurements of both the velocity of light and the ratio of units have by no means shattered the belief in the agreement and even the identity of these two magnitudes which, to judge from their original physical meaning, would seem to have hardly anything in common with one another.* For a transparent isotropic medium differing from air through its dielectric " constant " K, and showing but a negligible difference in p, Maxwell's theory gave the velocity of light c/ >JK, where K is taken in the electrostatic system and is thus a pure number. The refractive index of the dielectric medium with respect to air should therefore be given by * Many numerical data, together with the bibliography of the subject up to 1907, will be found in Encyklop. d. mathem., Wiss., vol. v., part 3, p. 186 et seq. ; Leipzig, 1909. Interesting details will be found in Prof. Whittaker's precious History of the Theories of Aether and Electricity, chapter viii., Longmans, Green & Co., 1910. 1 * 4 THE ORIGIN OF THE In order to test this predicted relation, Maxwell quotes the only example of paraffin. For solid paraffin Gibson and Barclay have found K = 1-975. On the other hand Maxwell takes the values of the refractive index n found by Gladstone for liquid paraffin (at 54 and 57 C.) for the spectrum lines A, D, and H, from which he finds, by extrapolation, for infinitely long waves n = 1-422. He takes infinitely long waves in order to approach as well as possible the conditions of the slow processes (static or quasi-static) upon which the measurements of the dielectric coefficient K were based. Putting together the values thus obtained, n = 1-422 ~ = 1-405. Maxwell confesses that their difference is too great to be thrown on the experimental errors ; he does not doubt, how- ever, that if JK is not simply equal to the refractive index, yet it makes up its essential part. He expects a better agree- ment only when the grain structure of the medium in question will be taken into account. It is universally known that Maxwell's predictions have found a splendid corroboration a quarter of a century later,* in the famous experiments of Hertz who has not only con- firmed the existence of electromagnetic waves, but also verified the approximate equality of their velocity of propagation with * In 1889. Hertz's papers are reprinted in vol. ii. of his Gesammelte Werke, under the title, Untersuchungen iiber die Ausbreitung der elek- trischen Kraft ; Leipzig, 1892. English version in Miscellaneous Papers, translated by Jones and Schott. ELECTEOMAGNETIC THEORY OF LIGHT 5 that of light by measuring the length of stationary waves and by calculating, on the other hand, the period of his electric os- cillator by the well-known approximate formula T JLG. c The agreement was satisfactory ; a better one could, at any rate, not be expected, seeing that the self-induction and the capacity of the oscillator (L, C) entering into the above formula corresponded to quasi-stationary conditions while Hertz's oscillations were of a rather high frequency. More- over, it is well known that Hertz and his numerous followers have imitated, with short electromagnetic waves, almost all the fundamental optical experiments. 2. Advantages of the Electromagnetic over the Elastic Theory of Light. It will be well to acquaint the reader with certain conspic- uous advantages offered by the electromagnetic as compared with the "elastic " theory of light, i.e. the theory based upon the assumption of an elastically deformable aether. In doing so we shall by no means attempt to give here the com- plete history of the luminiferous aether, but shall content ourselves with sketching a certain fragment of that compli- cated and interesting history, viz. that concerning the question of longitudinal waves (which had at any cost to be got rid of) and of the so-called boundary conditions. With this aim in view it will be enough to start from Green's work leaving aside the earlier investigations of Fresnel, F. Neumann, and others. Green's * aether is a continuous elastic medium endowed * G. Green, On the Laivs of Reflexion and Refraction of Light at the Common Surface of two Non-crystallized Media, Cambridge Phil. Trans., 1838, reprinted in Mathematical Papers, pp. 245-69. (In this paper Green denotes our following n by B and U + ~wby A.) On the Propaga- 8 6 GREEN'S AETHER with a certain resistance against compression k, i.e. with a compressibility I/k, with an elasticity of shape or " rigidity " n, and, finally, with a density, or inertia for unit volume, p. In isotropic bodies or media k, p, n are ordinary scalars, while in crystals n has in different directions different values or is, in modern terminology, a linear vector operator ; k, p retain their scalar character. In such an aether, two kinds of waves can exist and propagate themselves independently of one another, viz. longitudinal or dilatational waves, with the velocity and transversal or distortional ones, with the. velocity in the case of isotropy. Owing, however, to the discovery of the phenomena of polarization by Malus, Arago, and Fresnel, the physicists have convinced themselves that the luminous oscillations must be purely transversal. In order, therefore, to harmonize the theory with experience it has been necessary to get rid of the longitudinal waves. At first sight it would seem that it is enough to simply assume that only the transversal oscillations of the aether, and not the longi- tudinal ones, affect the retina. This, however, \vould not solve the difficulty. In fact, it can easily be shown that in the process of refraction at the boundary of two optically different media, purely transversal vibrations of the aether in the first medium,'* penetrating into the second, would split into two wave trains : a transversal and a longitudinal one. tion of Light in Crystallized Media, Cambridge Phil. Trans., 1839, re- printed in Math. Pap., pp. 293-311. *To wit, those contained in the plane of incidence. THE LONGITUDINAL WAVES 7 The velocities of propagation of these refracted waves being different, the corresponding two rays would make with one another, in general, a non-evanescent angle ; each of these rays on emerging from the second into the first medium, through any other boundary, would again be split into two : one consisting of transversal and another of longitudinal vibra- tions, and the two rays of transversal vibrations thus origin- ated would then certainly be accessible to our senses. In short, we should have a peculiar phenomenon of double re- fraction of light in an isotropic body. No traces, however; of such a phenomenon have ever been found experimentally,. The longitudinal waves, therefore, had to be got rid of in a more radical way. Now, the mathematical investigation of the subject has shown that the superfluous dilatational wave in the second medium dies away almost completely within a few wave- lengths from the boundary surface if it is assumed either that (1) The velocity of propagation v' of the longitudinal vibra- tions is very large as compared with the velocity v of the transversal ones, or (2) That this ratio, v' : v, of the velocities is very small. As will be seen later on, William Thomson has established (1888) the physical admissibility of the second assumption. Green, however, was convinced that the first was the only possible assumption, since he did not see his way to admit a negative k, i.e. a negative compressibility : a body endowed with a negative k would, at the slightest difference between its own pressure and that of its surrounding medium, expand or shrink indefinitely. For Green, therefore, the lower limit of k has been k = 0, and consequently, the lower limit of the ratio of the two velocities, by [1] and [2], He has thus been compelled to adopt the first of the two 8 GREEN'S THEORY assumptions, viz. that the dilatational waves are propagated with a velocity which is enormously greater than that of the transversal ones, in other words, that the ratio of the coefficients k/n is very large. In this manner Green's aether has become similar to an almost incompressible jelly. Next, in order to obtain from the differential equations of motion of his aether and from the boundary conditions, the laws of reflection and refraction at the interface of two iso- tropic media 1 and 2, Green introduces the supplementary assumption that the rigidity of the aether in the two media is the same, while its density has different values, This gives for the intensities of the reflected and the refracted ray, in the case of incident vibrations normal to the plane of incidence, two formulae identical with Fresnel's formulae for light polarized in the plane of incidence which are notoriously in good agreement with the experimental facts. Thus far, however, Green makes no use of his assumption k/n equal to a large number ; for in the case in question the longitudinal waves do not enter into play. They reassert themselves only when the incident light oscillates in the plane of incidence. Now, the formulae which in the latter case follow from Green's theory, do not agree with the corresponding Fresnel- ian formulae and deviate very sensibly from experiment ; in fact, they give for light reflected under the "angle of polari- zation " an intensity which differs too much from zero.* This is a serious objection against Green's theory. The substitution, for Green's n 1 = n. 2 , p l =j= p 2 , of the opposite assumption of Neumann or of MacCullagh : * Fresnel's formula gives zero for that intensity, while actually but a certain minimum is observed under the "angle of polarization"; this minimum, however, although still observable, is very weak as compared with the intensity of the incident light, and is most likely due to a heterogeneous transition layer at the interface of the two media. MAcCULLAGH NEUMANN 9 Pi = P2> U l 4= W 2 with the retention of all the remaining points of Green's theory, does not help the matter. In fact, the investigations of W. Lorenz (1861) and of Lord Eayleigh (1871), based upon the last assumption, lead to a result which emphatically contradicts experience, viz. to a formula for the ratio of amplitudes of reflected and incident light which, in the case of but slightly differing refractive indices of the two media, can be written A r : Ai = const, sec 2 i . cos &i* where i is the angle of incidence. This ratio vanishes for i == 7T/8 and for i = 3ir/8 ; thus we should have two different angles of polarization a phenomenon which nobody has ever observed. It has been necessary, therefore, to return once more to Green's assumption ^ = w 2 , p 1 =|= p 2 , and to meet the reflection and refraction difficulties by modifying Green's theory in some other direction. This has been done by Sir William Thomson who has replaced Green's jelly by a kind of. foam aether which will be described presently. Thus far we have been concerned with isotropic media. In anisotropic media, viz. in optically biaxial crystals, Green's aether t had three principal rigidities, n lf w 2 , n s , a single scalar coefficient k (as in isotropic media), and a constant density p, the coefficient k being again very large as com- pared with each of the three rigidities. We know already that under such circumstances the velocity (v') of longitudinal waves is very great as compared with the velocity (v) of trans- versal ones. For the latter, Green's theory gives at once the universally known Fresnelian equation * W. Lorenz, Pogg. Ann., vol. cxiv., 1861, pp. 238-50; Lord Rayleigh, Phil. Mag. for August, 1871, see especially p. 93. t On the Propagation of Light in Crystallized Media, already quoted, p. 200. 10 CEYSTALLINE MEDIA 7 -2 72 7 'I ___!! , .__*_.. . _ * [3] v 1 - njp v' 1 - n 2 /p v 2 - n s /p a result which is in excellent agreement with experimental facts. (In this formula, Z lf Z 2 , Z 3 , are the direction cosines of the wave normal with respect to the principal rigidity axes.) Thus far the propagation of light in a crystalline medium. Difficulties, however, arise in connexion with the treatment of reflection and refraction at the surface of a crystal, in con- tact with, say, an isotropic medium. For optically uniaxial crystals (n. 2 = n 3 , n^ =j=w 2 ) one could, after all, accept Green's assumption according to which the principal rigidity of the crystal (n^ corresponding to its unique rigidity axis should be equal to the rigidity of the aether in the adjacent isotropic medium. In the case, however, of optically biaxial crystals, having three different rigidities n v n. 2 , n^ one could hardly privilege any one of them, i.e. put it equal to the aether rigid- ity in the adjacent medium. And if we wished to meet this difficulty by assuming that the densities of the aether in the crystal and the adjacent medium are equal to one another, the previous, undesirable result would reappear, viz. two different angles of polarization (as in the case of ordinary, non-crystalline reflection). Lord Rayleigh* attempts to improve this weak point of Green's theory by assuming that the aether within biaxial crystals moves so as if it had in three orthogonal directions three different principal densities, p lt p 2 , p a , and ordinary scalar elastic coefficients fc, n, independent of direction and equal for all media. In passing from one medium to another it is the aethereal density only which is changed. This theory, involving a peculiar dependence of the aether's inertia upon the direction of motion, is based upon reasonings concerning the mutual action of the aether and the molecules of ponder- able matter. And it is precisely that interaction which is supposed to be the source of those directional properties of * Phil. Mag. for June, 1871. GREEN AND RAYLEIGH 11 the aether's mass. Now, Rayleigh's differential equations give for the square of the velocity of propagation of all possible waves (without, thus far, the exclusion of the longitudinal ones) the cubic equation * 2 2 72 J , ^3 __ __ rj.1 "*" * - - - * __ F* - B/ Pl V* - B/p. 2 "" V* - B/p 3 - (A - B)V* where B = n, A k + 4%/3. Introducing here again, after Rayleigh, Green's original assumption A/B = oo , we have that is to say, for one of the three waves, viz. the longitudinal t V = v' = QO , as in Green's theory, and for the remaining two, transversal ones, the cubic equation Vlfi V/f* V/P> t?s - B/p, But this equation does not agree with Fresnel's equation [3] which is notoriously a faithful representation of the ex- perimental facts. Thus, Rayleigh's theory, in its turn, had to undergo further radical modifications. Let us return for another moment to the more general equation [4], We see at a glance that it will yield, with any degree of approximation, the required Fresnelian equation, (k 4\ or + - ), instead n 3/ of being very great is, on the contrary, very small, that is to say, if we decide in favour of the second of the above alterna- tives which was rejected by Green a priori. It is precisely this assumption (2), page 7, i.e. k 4 A : jB = - + 5 = a very small number, n o * Glazebrook, Phil. Mag. (5), vol. xxvi., p. 521. 12 THOMSON'S FOAM-AETHER which is the starting-point of Thomson's aether theory (1888).* It is true that previous authors have already used that assumption. Fresnel has ascribed to his aether now an evan- escent and now an infinite compressibility. Thomson, how- ever, was the first to prove the physical possibility of that 4 assumption. In fact (2) requires at any rate A/B<^, and therefore &<0, that is to say, a negative compressibility, and that was the reason why Green thought that the aether's 4. stability calls for A/B > .5. Thomson proves, however, that o this is by no means a necessary condition of the stability of the aether. To see this, let us assume, after Thomson, that the aether either occupies a limited region of space and is fixed at its boundary, or extends indefinitely in all directions but that the displacements, , etc., of its particles decrease in such a manner that the products dfix, etc., become infinitesimals at least of the third order. Then the expression for the work to be done upon the aether in deforming it infinitesimally from its natural (or neutral) state can, by partial integration, be reduced to the form W = fftAo* + ZB^\dr . . . [5J where o- is the dilatation, or the div of the displacement, and 2 = K.^ = an ordinary scalar 7T. The last two equations will be shortly referred to as the solenoidal conditions. A glance upon (!') suffices to show the waste of paper (and of time) involved in the Cartesian expansion of formulae, (1) in the present case, which are, by their intrinsic nature, vectorial.* As to the immediate general consequences of (1), it will be enough to mention those concerning the electromagnetic energy and its flux. The density of energy is, in the general case of a crystalline medium, given by -W) . . . (2) Now multiplying (scalarly) the first of (1) by E, the second by M, and adding both, we have 1 du cdf = E curl M - M . curl E = - div VEM. Thus the flux of energy, per unit time and unit area, is seen to be F = c VEM . (3) * The Cartesian splitting will be avoided as much as possible. Those readers who are not familiar with vectors can acquire whatever is necessary to follow the present deductions by reading chapter i. of the author's Vectorial Mechanics, Macmillan, 1913. 2 18 PLANE WAVES < This vector, c times the vector product of E and M, is uni- versally known as the Poynting vector. It is this vector which, by its direction, defines the optical ray, in an isotropic medium as well as in a crystal. If E points upward and M to the right, the flux, and, therefore, the ray is directed forward. The intensity of light at a given point is measured by the time average of the density u of electromagnetic energy. As a preparation for the following sections we need only those integrals of the equations (1) which correspond to plane waves. Let the unit vector n be the wave-normal, and let the scalar distance s be measured along n. Then, by the very definition of plane waves, the vectors E, M, depend only on s and the time t. Under these circumstances the Hamiltoniari V becomes and, therefore, for any vector R, curl R = V V R = Vn OS and div R = vR = n -. OS Thus, for any plane waves, equations (1) become : = V = - Vn 7 ^i '' W c It ^s J or, since n is constant in space, PLANE WAVES 19 1 VnM ; 1 >* = VEn, '. (4a) <)S C <> <)S In what follows we shall be concerned with monochromatic light, i.e. with simple periodic waves. Then E, M are pro- portional to e im(s - vt) , where i = ,J - 1, m = const. ( = 2?r divided Jby the wave-length in the given medium), and v the velocity of propagation, so that is the refractive index of the medium. Thus 3 3 = - imv, = im, M ds and the equations (4a) become, independently of m, ?#E = VMn; v - M = VnE. c c The solenoidal conditions (46), which now require A'En = Mn = 0, are already satisfied,* since nVMn = nVnE = 0, identically. Let D be the dielectric displacement, i.e. D = KE. Then the last pair of equations will become -D = VMn; ~M = VnE (5) c c These will be our fundamental equations, valid for plane waves of wave-normal n, in any homogeneous medium, be it * The trivial case v = being, of course, disregarded. 2 * 20 LIGHT- VECTOES AND RAY isotropic or crystalline. They contain, as a consequence, the solenoidal conditions Dn = 0, Mn = 0, whose plain meaning is : the magnetic force and the dielectric displacement are perpendicular to the wave-normal, i.e. are contained in the plane of the wave. In other words, M and D (not E, in general) are purely transversal. Again, multiplying the first of (5) by M, and the second by E, we have MD = EM = 0, i.e. M j_ E, D. But, in general, En ^ 0. We shall return to the general relations (5) when we come to treat, in detail, the optical properties of crystals. For the present let us confine ourselves to isotropic media. Then D coincides with E in direction, being simply K times E. Thus, in isotropic media, we have not only E j_ M and M _L n, but also E _L M. In short, E, M, n, are all mutually perpendicular. Again, multiplying, say, the second of (5) by M, we have V -M* = MVnE = nVEM . (6) c or also, by (3), vM' 2 = nF (6a) From (6) we see that nVEM is always positive, i.e. that E, M, n is a right-handed system of orthogonal vectors. And since F is concurrent with n, E, M, ray is also a right-handed system. ENERGY AND ITS FLUX 21 Further, multiplying the first of (5) by E, and remembering that D = KE, we have -KE* = nVEM = -M*. c c Thus KE* = M*, . , .. , . . (7) i.e. half of the energy is electric, and half magnetic. Waves satisfying the latter condition together with E j_ M are called pure waves. The density of electromagnetic energy, ^KE 2 + M' 2 , now becomes u = KE* = M* . . ' , . (8) In order to obtain the velocity of propagation v, elimin- ate from (5) either M or E. Thus, remembering that n 2 = 1, = VnVEn = E - (En)n, and since En = 0, the well-known result. Finally, the flux of energy can be written, again by the second of equations (5), r 2 r 2 BJ2 F = C -VEVnE = n, V V i.e. by (8) and (9), F = u:vn = tiv, -.' . , v (10) where Y is the velocity of propagation in magnitude and direction. This simple formula can be read : the electromagnetic 22 BEFLECTION AND EEFEACTION energy is carried forward with the vector velocity Y, as if it were a fluid of density u. 4. Reflection and Refraction at the Boundary of Isotropic Media; E in Plane of Incidence. Since any vibrations of the electric vector E (and of its magnetic companion) can always be split into two rectilinear vibrations in two mutually perpendicular directions, it is possible, and convenient, to treat separately the two cases of monochromatic plane waves of rectilinearly polarized incident light : 1st, E parallel, and 2nd, E perpendicular to the plane of incidence. As concerns M we know already that it is en- tirely determined by E and by the direction of the ray, its intensity being given by M' 2 = KE' 2 , and its direction by the circumstance that, in each of the adjacent media, E, M, ray is a right-handed orthogonal system of vectors. Let us begin with the first case, E in plane of incidence. Let the interface of the two isotropic media, whose permit- tivities (corresponding to the given frequency) are K and K', be a plane. Take this as the y, z plane, and let the normal to the interface drawn towards the first medium (K) be the axis of positive x. Let the plane waves arrive from the first medium (K) towards the second (K'). It will be convenient to take the 2-axis along the intersection of one of the wave planes with the boundary. Then x, y will be the plane of incidence. Denoting the angle of incidence by a, the angle of reflection by Oj, and the angle of refraction by ft (Fig. 1), use the abbreviations a = - cos a, a l = cos a 1? a' = - cos ft 1) = sin a, 6 X = sin a l5 b' sin ft, Then the distances measured along the incident, the reflected, and the refracted rays will be E IN PLANE OF INCIDENCE ax -f by, a^x + b^, a'x + b'y, 23 respectively. If, therefore, v and v' be the velocities of pro- pagation in the first and the second medium, i.e. by (9), then E, M will be simple periodic functions of the arguments ax + by - vt for the incident, a-^x + b-$ - vt for the reflected, and a'x + b'y vt for the refracted wave. Thus, g and g' being constant factors, the vectors E and M will be propor- tional to the exponential functions exp. ig(< exp. ig(a l x + b^y - exp. ig'(a'x + by - vt) in b^y - vt) b'y - v't) the incident rayl ,, reflected ray refracted rayj . (11) K FIG. 1. The electric force being in the plane of incidence (plane of Fig. 1), the magnetic force will be parallel to the axis of z. Let its intensity, taken positive along the positive z-axis, be denoted by M for the incident, by M l for the reflected, and by M' for the refracted wave. Let E, E v E' be the correspond- ing symbols for the electric forces, taken positive in the direc- tion of the arrows. Then, by the second of equations (5), 24 BOUNDARY CONDITIONS E = V M, E, = -M lt E' = V M' (12) c c c The differential equations (1) being now all satisfied, for each wave separately, it remains only to take account of the boundary conditions. These are, in virtue of the equations (1) themselves, as is well known : 1st, the continuity of the whole magnetic force M,* 2nd, the continuity of the tangential component of the electric force E, and 3rd, the continuity of the normal component of the dielectric displacement D = KE. Thus we have, for x = 0, M+M 1 = M' t ....... (a) E cos a - E 1 cos a x = E' cos ft . . . (b) K(E sin a + E l sin 04) = K'E' sin ft . . (c) Since these conditions are to be satisfied for every y and for all times t, we have in the first place, by (11), prior to any considerations concerning the amplitudes, (jb = gb l = g'b', gv = g'v', or, remembering the meaning of b, etc., . a i = a ' \ . (13) sin a : sin p = v : v = n) i.e. the two fundamental laws of geometrical optics : equality of angles and identity of planes of incidence and of reflection, and Snellius law of refraction. At the same time we have, by (9), for the ratio n, independent of a, i.e. for what is called the refractive index of the second medium relatively to the first, * Since /i = p = 1. FRESNEL'S FORMULAE 25 In virtue of (12), the third boundary condition (c) now becomes identical with (a). Thus but two conditions are left which, by (12), can be written entirely in terms of the electric forces, E + E l = -, E' . . . . (a) v COS a Eliminate E'. Then ''sin ft cos a\ cos a sin ft cos 8 sin a E^ /sn ft cos a\ _ E \sin a cos ft) "~ Add (a) to (ft) and use the above refraction law, (13). Then the result will be E' /sin a cos ft\ E \sin ft cos a/ =2. After an easy trigonometrical transformation we have . (15) E l tan (a - ft) E E' E tan (a + ft) 2 cos a sin /? sin (a + ft) . cos (a - 0) These are the famous formulae of Fresnel for incident light whose "plane of polarization " is normal to the plane of inci- dence. According to Fresnel the vibrations of the elastic aether were perpendicular to the plane of polarization. In the present case, therefore, the Fresnelian vibrations would be parallel to the plane of incidence. On the other hand, 26 FKESNEL OK NEUMANN according to Neumann the aether vibrated in the plane of polarization. We thus see that Fresnel's light-vector* corresponds to E (or D), Neumann's light-vector corresponds to M. The distinction between E and D remains immaterial until we come to treat anisotropic media. The electromagnetic theory of light has thus united in itself, as far at least as reflection and refraction are con- cerned, both theories, Fresnel's and Neumann's, and the famous quarrel about the direction of sethereal vibrations with respect to the plane of polarization has, in its original sense, become an idle controversy. The problem has now acquired a different meaning, viz. : is the action of light to be ascribed to the electrical or to the (inseparable) magnetic oscillations ? The experiments of O. Wiener t on stationary light waves are supposed to speak in favour of the electric vector, as far at least as the action upon photographic plates or the excitation of fluorescence are concerned. The ratio of E l to E has, by (15), a positive value. Since, however, the positive senses of E, E 1 have been taken along the arrows of Fig. 1, the positive sign of the ratio signifies that the tangential component of the electric force under- goes a change of phase by 180 on being reflected. The phase of the normal component is not changed. Also the refracted ray proceeds without change of phase. The intensity of light being measured by the time-average of u or of KE 2 = M 2 , and the oscillations being periodic, the intensities I, I lt I' will be given by the squared amplitudes of E, JBJj, E' multiplied by K, K v K' respectively, i.e. ' In the case of normal incidence, i.e. for a = ft = 0, we have * That is, the vibrating or periodically variable directed magnitude. t Started in 1890. See Ann. der Physik, vol. xl., p. 203. BEEWSTER'S LAW 27 from (15), or simpler, returning to the original equations (a), (b) which now become E + E l = nE' ; E - E l = E', E n - 1 E' 2 Notice that in this case EI + E' = E, an obvious property. Returning to the first of the general formulae (15) we see that for a + ft = 7T/2 E l = 0, i.e. light polarized normally to the plane of incidence is not reflected at all. A glance upon Fig. 1 suffices to see that this happens when the reflected ray would be perpendi- cular to the refracted one. This is Brewster's law. The corresponding angle of incidence a , called the angle of polarization, is determined by sin a ft sin a rt Yl = " = sin # . /TT sin U - a Thus tan a = n .. . . (17) It can be easily verified that the light incident at the angle of polarization penetrates entirely into the second medium, i.e. that I' = I. In fact, for a = a , the second of (15) be- comes & = 2 cos 2 a . cot a _ 1, E COS (2a - -TT) n and therefore, by (16), I'll = in agreement with the principle of conservation of energy. 28 EEFLECTION AND EEFKACTION It may be well to mention that Fresnel's formula E t : E = tan (a - ft) : tan (a + /3), showing in general a very good agreement with observation, deviates somewhat from reality in the neighbourhood of the angle of polarization. It was found that these slight deviations are influenced by external circumstances * which produce at the reflecting face a thin optically heterogeneous sheet. In fact, it has been possible to account for these deviations theoretically by assuming such a transitional layer. 5. Reflection and Refraction ; E _L Plane of Incidence. Note on the Transition Layer. In this case the electric vector is parallel to the boundary of the two media, i.e. to the axis of z, while the magnetic vector is contained in the plane of incidence. In the habitual terminology, the incident light is polarized in the plane of polarization. Proceeding as before, i.e. writing down the boundary con- ditions which now require the continuity of the whole electric force and of the whole magnetic force, we get again a x = a, sin a : sin p = n, and for the ratios of the electric amplitudes, EI sin (a - E ~ sin (a + ft) K 2 cos a sin p E ' sin (a + P These formulae are again identical with those known as Fresnel's formulae for incident light polarised in the plane of incidence. The electric force again corresponds to Fresnel's light-vector. *Thus, for instance, according to Drude, the deviations from Fresnel's formula for a freshly broken face of a crystal of rock-salt were very small, but, the reflecting face being exposed to the action of -air, the deviations began to increase rapidly. E NOEMAL TO INCIDENCE PLANE 29 For normal incidence formulae (18) become, in appearance only, indeterminate. Keturning to the original form of the boundary conditions, the reader will obtain for a = 0, with- out trouble, - 1 - n ~- EI 2 From the first of (18) we see that EJE never vanishes, i.e. that light polarized in the plane of incidence is not extin- guished at any angle of reflection. In the preceding section we saw that light polarized perpendicularly to the plane of incidence is not reflected at all when a becomes equal to the polarization angle a = arc tan n. Thus, common or natural light (which can be considered as consisting of both the above kinds), when reflected under the angle a , would be polarized in the plane of incidence, i.e. so that only the electric oscillations, normal to the plane of incidence, would remain. According to the above Fresnelian formulas this polarization would be complete, while experiments give a slight residue of electric vibrations contained in the plane of incidence. These small deviations of theory from observation can, however, as was already remarked, be accounted for by assuming a hetero- geneous layer of transition, i.e. having a permittivity K which in the narrow limits X = tO iC = depends upon x, and assumes for x < - e and x > e the constant values K' and K respectively. Drude * has shown that an approximate treatment, in which the integral values * Cf. Drude's Lehrbuch der Optik, second edition, Leipzig, 1906, pp. 272-80, or English translation by Mann and Millikan, 1913, Longmans, Green & Co. 30 TRANSITION LAYER dx , \' J are the only relevant ones, is practically sufficient. If the ray, incident under the angle a = arc tan n, is rectilinearly polarized in a plane oblique to the plane of incidence, say under the azimuth of 45, then the reflected light, instead of being polarized rectilinearly contains also traces of elliptic * polarization, in agreement with observation. The ellipse re- sulting from Drude's theoretical investigation is very long, the ratio p of the minor to the major axis being (for = 45) * JK + K p ~ x TT-~r J _ (K - K) (K - K') ^ where X is the wave-length (in vacuo). The absolute value of the integral does not exceed that which would correspond to K = ,JKK' = const. Thus we have from (19), for the upper limit of the thickness I = 2e of the transition layer, I 1 (20) .. A. TT Vl + n * n - 1 Now, the observed value of p for heavy flint glass in con- tact with air (n = T75 for sodium light) is P = 0-03, this being the highest ratio of axes yet observed. In this case formula (20) would give Z/X = 0-0174, i.e. I = 1-023 . 10~ 5 mlm. In other cases we would have obtained thicknesses even a hundred times smaller. In fact, for other kinds of glass, of smaller refractive index, values of p have been found scarcely exceeding 0'007, and Lord Rayleigh's value for water, whose * The ellipticity being due to the difference of phase of the component oscillations. TOTAL EEFLECTION 31 surface has been carefully cleaned, was as small as 0-00035. The corresponding thickness of the layer of transition would be nearly as small as 10~ 7 mlm., that is, of the order of mole- cular dimensions. 6. Total Reflection. Let the " first " medium, from which the light arrives, be optically denser than the " second," i.e. in the above symbols, Then, for a = a> = arc sin n, sn 0= andfora>w si That is, ft will cease to be a real angle. Thus for a > a>, a refracted wave, whose amplitude, however, in penetrating the second, thinner medium, will die away exponentially, the more rapidly the greater the difference a o>. In fact, returning to (11), and remembering that a = cos {3, b' = - sin /?, notice that the forces in the refracted ray are proportional to e - ig'(x cos /3 + y sin /3 + v't) ^ where cos ft = 1 1 - sm * a \ n 2 is imaginary (since sin a>w). Write, therefore, cos ft = *y> 32 TOTAL KEFLECTION where y / S1I V 1 is real. Thus the part of the above exponential depending on x will become eve' x . The minus sign is, by physical reasons, to be rejected, since it would give an indefinitely increasing amplitude of waves pene- trating into the thinner medium (x negative). What re- mains, therefore, is where x' = - x. If T be the period of oscillations and A/ the wave-length in the thinner medium, g'v' = STT/T, g' = 27T/A', so that the above damping factor becomes Thus, if a - o> is sensibly positive, the amplitude of the re- fracted vibrations will become evanescent in a depth of a few wave-lengths. Notice that besides this factor we have only e -ig'(y sin/3 + v't)^ so that these rapidly damped oscillations are propagated in the second medium along the interface, viz. in the direction of the negative 7/-axis. Let us now see what the properties of the reflected waves are, after the "limiting angle" a = w = arc sin n has been exceeded. Take, for example, the case in which the incident light is rectilinearly polarized under the angle = 45 relatively to the plane of incidence. Let P be the component of E in the plane of incidence, and Z its component along the -axis. Let Pj, Z l have similar meanings for the reflected wave. Thus P l and Z l will be what has been denoted by E l in (15) TOTAL EEFLECTION 33 and in (18) respectively, and since (for = 45) P = Z, we shall have PI _ _ tan (a - ft) sin (a + ft) = _ cos (a + P) ^ tan (a + 0) ' sin (a - ft) ~ cos (a - ft)' Since sin ft > 1, this ratio will have a complex value, say pe iA where p and A are real. The former will be the ratio of the absolute values [PJ, \Z^\ or the ratio of the amplitudes, and the latter the phase difference of the component con- tained in the plane of incidence and the component normal to this plane. Developing trigonometrically the last expres- sion, remembering that sin ft = - sin a, and that 1 cos ft = iy, y = Jsin 2 a - ri 1 , . . (21) VI the reader will find , _ i 06 ~ (sin 2 a - i . ny cos a) 2 sin 4 a + W 2 y 2 COS 2 a ' whence p = 1, i.e. |Pj| = \Z, , . . (22) and using (21), A oos^/sin^^^ 2 sin 2 a Thus the amplitudes of the reflected components P lf ^ will be equal to one another,* while their phases will differ *The reader will easily show that these reflected amplitudes are equal to the incident ones, i.e. (whether is 45 or not) ampl. P l = ampl. P, ampl. Z x = ampl. Z. This result could be expected, without returning to (15) and (18). In 3 34 TOTAL [REFLECTION from one another, the more so the more the limit angle a>, will be ellipti- cally polarized. This is a general property of total reflection which in its essence has already been mastered by Fresnel. If 4=45, then, instead of p = 1, while the phase difference continues to obey the formula (23). The azimuth 45 is of particular interest because it gives for A = 90 a circularly polarized reflected ray. In order to obtain A = 45 we have to make, by (23), COS a sin 2 a - ri* = tan 22-5, that is, for glass of refractive index - = 1*51, say, in contact with air, either a = 48 37' or a = 54 37'. By two reflections under any of these angles * the phase difference A = , i.e. circularly polarized light is obtained. This is the principle of the universally known glass rhomboe'der of Fresnel (a == 54). 7T Notice that by a single reflection the phase difference of ^ could not be obtained since, for - = 1*51, the maximum of A n amounts, by (23), only to 45 36'. fact, it is enough, to remember that in the thinner medium (x , as is necessary for total reflection ; in fact, o> = arc sin - - - = 41 26'. 1*51 CRYSTALLINE MEDIA 35 7. Optics of Crystalline Media : General Formulae and Theorems. The propagation of plane waves of monochromatic light in a crystalline medium obeys the formulae (5) deduced in Section 3, ?D = VMn; ^M^VnE . > . (5) c c Here n is the wave-normal, a unit vector, and v the velocity of propagation of the waves. The dielectric displacement D = KB is a linear vector function of the electric force E. The whole optics of a homogeneous crystal is contained in the two simple equations (5), the properties of the crystal being denned by the vector operator K. As was already mentioned, formulae (5) contain the funda- mental relations Mn = 0, Dn = 0, i.e. the magnetic force and the dielectric displacement lie in the wave-plane, or are purely transversal. Notice that what corresponds to Fresnel's light-vector is the dielectric displace- ment D, and not the electric force E. The latter is in general not contained in the wave-plane, i.e. En =1=0. Formulas (5) give also the fundamental relations MD = 0, ME = 0. Thus the magnetic force is always normal to both the electric force and the displacement. The ray, denned by the direction of the energy flux P = 36 PEOPAGATION IN CBYSTALS is always normal to E, M, but in general oblique relatively to the wave-plane. Multiplying the first of (5) by E, and remembering that EVMn = MVnE, we have ED = M* . . . . (24) of which equation (7) is but a special case. Thus the lumin- ous energy, in a crystal as well as in an isotropic medium, consists in equal parts of electric and of magnetic energy. In order to obtain an equation for the velocity of propaga- tion v as dependent upon the direction of the wave-normal n, eliminate from (5) the magnetic force. Then, remember- ing that VAVBC = B(CA) - C(AB), and that n 2 = 1, ^ D = VnVEn = E - (En)n, c and since D = KB, where the bracketed expression, as well as K itself, is a linear vector operator. The last equation will conveniently be written D = KE = (En) n, . . (25) where the bracketed expression is again a linear vector operator, having the same principal axes as K and the cor- responding principal values Z a c 2 c 2 c 2 K 2 , K 6 being the principal values of K. FEESNEL'S EQUATION 37 Now, Dn = 0. Thus multiplying (25) scalarly by n and remembering that the scalar product (En) is in general different from zero, we have, for the velocity of propagation v, the equation = ' ' ' ' : (26) which is but the vectorial form of the famous Fresnelian equation. In fact, denote by n v w 2 , n s the components of the wave-normal n, or its direction cosines, with respect to the principal electrical axes of the crystal, and use the abbrevia- tions Then the Cartesian expansion of (26) will be = (26a) v%- -i'- 1/3- which is Fresnel's equation. For any given direction of the normal n we have, from (26) or (26a), in general two different absolute values of v, say v' and v". Let the vectors E, D corresponding to v' and to v" be E', D' and E", D", respectively. Then, by (25), If, therefore, the waves are to be propagated with the same velocity (either v' or v"), i.e. if they are not to split into two trains of waves, then the vector D must have one of the two directions or 8 38 DIRECTION OF VIBRATIONS where the + signs, being, of course, irrelevant, have been omitted. These two privileged directions s' and s" of light oscillations, belonging to a given normal n, are perpendicular to one another. In fact, denoting the vector operator in (26) by w, we can write Fresnel's equation no>n = 0, . . . . (266) and, instead of (27), s' = a/n, s" = to"n, and since V = CD CD V* and, therefore, s's" = V* - V By (266) both terms in the numerator vanish. Thus, if only v' =j= v", s's" = 0, .... (28) that is, s' _L S". Q.E.D. The case v' = v", which occurs only for certain directions of the normal n, will be considered a little later on. If the vector D has neither of these two privileged direc- * For so is also the permittiyity-operator K. DOUBLE EEFEACTION 3$ tions, then the waves are split into two separate trains. In this case, D being purely transversal, it can always be split into two components, D' along s' and D" along s". One train of waves will carry away with the velocity v' the com- ponent Z)' of the dielectric displacement, and another, with velocity v", the component D". It is this which constitutes what is called double refraction in crystals. The usual form of the equation for the direction cosines of Fresnel's light-vector can easily be obtained from (27). In fact, writing summarily s for s' or s" and denoting by s v s 2 > % the components of s along the principal electrical axes of the crystal, we have l 2 3 whence the required relation 4 K* - 'V) + v 2 W - ^ + - ^ - "o 2 ) " ; < 29 ) s l S 2 6 3 Si, etc., are proportional to the direction cosines of the vector D which corresponds to Fresnel's light-vector. In order to determine the direction of the ray, i.e. that of the energy flux F = oVEM, return once more to the equations (5). The second of these gives F = - VEVnE = - [jEJ'n - (En)E] ; 'V V now, by (25), writing again o> = [v 2 - C 2 JK] - 1 , E= - c 2 (En)-n; . 'V- " . (30) j\ thus, 40 EAY AND WAVE NORMAL If, therefore, p is a unit vector along the luminous ray, and cr a scalar to be determined presently. In order to obtain the value of ), ^if t -4 < ' -t (35) v" = v., v lt vJ Thus, v 1 c/ JK^ v 2 = c/ V-^2 v z = c l \/^3 are tne 80 " called principal velocities of propagation. If the normal is along the first electrical axis, the waves are propagated with the second or the third principal velocity (according to the direction of D), and so on, by cyclic permutation. In order to determine the directions s', s" in which D must 42 ELECTEICAL AXES vibrate in each of the above three cases if the wave is not to split, return * to the equation I D = E - (En)n, c already obtained from (5), p. 36. This gives, for n = i. c 2 and, therefore, by (35), But D' = JE" = K^ii + K 2 E 2 'j + # 3 # 3 'k. Thus EJ = 0, and since K 2 =!= J 3 , D' ~tr 77 '\ A 2 ^ 2 J* Similarly we shall find D" = K 3 E s "Vi. Analogous relations will take place for n = j, k. Thus, denoting by || the paral- lelism of vectors, we have for n = (35a) n = i, j, k^ D' II j, k, i . D' II k, i, j J These relations, together with (35), can easily be expressed in words. In passing we have also seen that the electric force E, being in general oblique, becomes for n = i, j, k, purely transversal, i.e. falls into the wave-plane. And since M is always in the wave-plane, the ray p coincides with the normal n when this coincides with any of the principal electrical axes of the crystal. * Since (27) or (25) becomes in the present case indeterminate owing to the fact that En will now vanish. BIAXIAL CEYSTALS 43 9. Optical Axes. Beturning to Fresnel's equation (26a) for the velocity of propagation, use the abbreviations (36) 7 = Then V> (37) Let our previous v correspond to the upper, and v" to the lower sign of the square root. Let us find those particular directions of the wave-normal n for which the two velocities of propagation become equal, v' = v". The necessary and sufficient condition for this equality is (a -f J3 - y) 2 - aj3 = 0. But, by (34), a is positive, ft negative, and 7 positive. Thus, we must have, separately, a + ft - y = 0, aft = 0. Now a = is inadmissible ; for then /? would be equal 7 while these magnitudes have opposite signs. The only possibility is, therefore, /? = 0, a = 7, i.e. n. 2 = 0, V(V ~ V) = ' w s 2 K 2 - V)- 44 BIAXIAL CRYSTALS But n 2 = 1, so that n./ = 1 - n^. Ultimately, therefore, the required directions of the normal are = Ml^ \ ^ 2 - V (38) These particular directions are called the optical axes of the crystal. One axis is obtained by taking in (38) the signs -f + (or - - ), the other axis by taking + (or + ). Thus, in the most general case, for different K v K 2 , K s , and therefore for different v v v 2 , v 3 , the crystal has two optical axes contained in the plane i, k, that is to say, in the plane of those two electrical axes to which correspond the FIG. 2. smallest and the greatest principal values of the permittivity- operator K. The orientation of the optical axes is sym- metrical with respect to the electrical axes i and k. Substituting (38) in (37), we have v' = v" = f 2 - * - . (39) This then is the common value of the two velocities when the wave-normal n coincides with one of the optical axes. It is easy to show that in this case the light-vector D can have any direction in the wave-plane. Independently of the orientation UNIAXIAL CKYSTALS 45 of D the velocity of the wave is, in these circumstances, always equal v%. If y p y 2 be the angles which any wave-normal n makes with the two optical axes A x , A 2 (as in Fig. 2), formulaB (37) for the corresponding velocities of propagation can be written ^7s *f v i 2 "^" V 3 2 "*~ ( v i 2 "~ v a 2 ) * cos ^1 + Ya) (^^) v2 10. Uniaxial Crystals. If the electrical properties of the crystal are axially sym- metric, say, with respect to the axis k, in other words, if K = K and therefore then (38) becomes HJ = 0, n 2 = 0, w 3 = 1. That is to say, the two optical axes coincide with the axis of electrical symmetry to which corresponds the principal per- mittivity K s , i.e. the velocity v s . The crystal is then uniaxial. Since, in the present case, y l = y 2 = y, say, so that cos y = n s , the two velocities of propagation corresponding to any given direction of wave-normal n, making with the optical axis the angle y, are, by (40), y =. vt'i COS . (41) y + v./ sm j y ) The velocity i/, which is constant, corresponds to what is called the ordinary, and v" depending upon the angle y corresponds to the extraordinary wave. To these waves in which the light-vector D has one of the orthogonal directions 46 UNIAXIAL CEYSTALS S', 8" correspond, in general, two different rays, the ordinary ray p', and the extraordinary ray p". A discussion of further details would not answer the pur- poses of this little volume. The reader will find them in every work on optics, whether based upon the electro- magnetic or upon the older theory. INDEX. T}ie numbers refer to the pages. , 5-15. Angle of polarization, 8, 27. Arago, 6. Axes, electrical, 37, 41. - optical, 44. BIAXIAL crystals, 44. Boundary conditions, 7-9, 24. Brewster's law, 27. CIRCULAR polarization, 34. Compressibility of aether, 6. DENSITY of aether, 6, 8. Dielectric displacement, 19, 35. Double refraction, 39. Drude, 28, 29. ELASTIC theory of light, 5-13. Electric force, 16, 35. Electromagnetic system, 1. Electrostatic system, 1. Elliptic polarization, 30, 34. Energy, 17. Extraordinary wave and ray, 45, 46. FIZEAU, 3. Foucault, 3. Flux of Energy, 17, 21, 35. Foam-sether, 13. Fresnel, 6, 26. Fresnel's equation, 10, 37. formulae, 25, 28. glass rhomboeder, 34. GIBSON and Barclay, 4. Gladstone, 4. Green, 5-10. HAMILTON, 1. Heaviside, 16. Hertz, 4-5, 16. INDEX, refractive, 3, 4. Intensity of light, 18, 26. KOHLBAUSCH, 2. LAPLACIAN, 1. Light-vector, 26, 35, 39. Longitudinal and transversal waves, 6-7. Lorenz, 9. MACCULLAGH, 8. Magnetic force, 16, 35. Malus, 6. Maxwell, 1-5, 16. Maxwell's equations, 16, 17. NEUMANN, 8, 26. Normal, wave-, 18. ORDINARY wave and ray, 45, 46. PERMEABILITY, 1, 14-15. Permittivity, K, 16. Plane-waves, 18-19. Polarization angle, 27. 47 INDEX Poynting vector, 18. Pure waves, 21. RAY, 18, 20i 35, 40, 45. Rayleigh, 9, 10-11, 13, 30. Reflection and refraction, 22-34. Rigidity of aether, 6. SNELUUS' law, 24. Solenoidal conditions, 17. Specie inductive capacity, K, 1, -13, 14-15. Stability, of aether, 12. THOMSON, W., 2, 7, 9, 12. Total reflection, 31-4. Transition layer, 28, 29 31. Transversal vibrations, 14, 20, 35. UNIAXIAL crystals, 45. Units, ratio of, 1. VECTOR operator, 15, 16. Vector potential, 1. Velocity of propagation, 2, 21, 41. Voigt, 2. WEBER, 2. Wiener, 26. Whittaker, 3. PRINTED IN GREAT BRITAIN BY THK UNIVERSITY PRESS, ABERDEEN. 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