Chapter 21

Bibliography.—There are numerous text-books which give elementary expositions of light and optical phenomena. More advanced works, which deal with the subject experimentally and mathematically, are A. B. Bassett,Treatise on Physical Optics(1892); Thomas Preston,Theory of Light, 2nd ed. by C. F. Joly (1901); R. W. Wood,Physical Optics(1905), which contains expositions on the electromagnetic theory, and treats “dispersion” in great detail. Treatises more particularly theoretical are James Walker,Analytical Theory of Light(1904); A. Schuster,Theory of Optics(1904); P. Drude,Theory of Optics, Eng. trans. by C. R. Mann and R. A. Millikan (1902). General treatises of exceptional merit are A. Winkelmann,Handbuch der Physik, vol. vi. “Optik” (1904); and E. Mascart,Traité d’optique(1889-1893); M. E. Verdet,Leçons d’optique physique(1869, 1872) is also a valuable work. Geometrical optics is treated in R. S. Heath,Geometrical Optics(2nd ed., 1898); H. A. Herman,Treatise on Geometrical Optics(1900). Applied optics, particularly with regard to the theory of optical instruments, is treated in H. D. Taylor,A System of Applied Optics(1906); E. T. Whittaker,The Theory of Optical Instruments(1907); in the publications of the scientific staff of the Zeiss works at Jena:Die Theorie der optischen Instrumente, vol. i. “Die Bilderzeugung in optischen Instrumenten” (1904); in S. Czapski,Theorie der optischen Instrumente, 2nd ed. by O. Eppenstein (1904); and in A. Steinheil and E. Voit,Handbuch der angewandten Optik(1901). The mathematical theory of general optics receives historical and modern treatment in theEncyklopädie der mathematischen Wissenschaften(Leipzig). Meteorological optics is fully treated in J. Pernter,Meteorologische Optik; and physiological optics in H. v Helmholtz,Handbuch der physiologischen Optik(1896) and in A. Koenig,Gesammelte Abhandlungen zur physiologischen Optik(1903).The history of the subject may be studied in J. C. Poggendorff,Geschichte der Physik(1879); F. Rosenberger,Die Geschichte der Physik(1882-1890); E. Gerland and F. Traumüller,Geschichte der physikalischen Experimentierkunst(1899); reference may also be made to Joseph Priestley,History and Present State of Discoveries relating to Vision, Light and Colours(1772), German translation by G. S. Klügel (Leipzig, 1775). Original memoirs are available in many cases in their author’s “collected works,”e.g.Huygens, Young, Fresnel, Hamilton, Cauchy, Rowland, Clerk Maxwell, Stokes (and also hisBurnett Lectures on Light), Kelvin (and also hisBaltimore Lectures, 1904) and Lord Rayleigh. Newton’sOpticksforms volumes 96 and 97 of Ostwald’s Klassiker; Huygens’Über d. Licht(1678), vol. 20, and Kepler’sDioptrice(1611), vol. 144 of the same series.Contemporary progress is reported in current scientific journals,e.g.theTransactionsandProceedingsof the Royal Society, and of the Physical Society (London), thePhilosophical Magazine(London), thePhysical Review(New York, 1893 seq.) and in theBritish Association Reports; in theAnnales de chimie et de physique and Journal de physique(Paris); and in thePhysikalische Zeitschrift(Leipzig) and theAnnalen der Physik und Chemie(since 1900:Annalen der Physik) (Leipzig).

The history of the subject may be studied in J. C. Poggendorff,Geschichte der Physik(1879); F. Rosenberger,Die Geschichte der Physik(1882-1890); E. Gerland and F. Traumüller,Geschichte der physikalischen Experimentierkunst(1899); reference may also be made to Joseph Priestley,History and Present State of Discoveries relating to Vision, Light and Colours(1772), German translation by G. S. Klügel (Leipzig, 1775). Original memoirs are available in many cases in their author’s “collected works,”e.g.Huygens, Young, Fresnel, Hamilton, Cauchy, Rowland, Clerk Maxwell, Stokes (and also hisBurnett Lectures on Light), Kelvin (and also hisBaltimore Lectures, 1904) and Lord Rayleigh. Newton’sOpticksforms volumes 96 and 97 of Ostwald’s Klassiker; Huygens’Über d. Licht(1678), vol. 20, and Kepler’sDioptrice(1611), vol. 144 of the same series.

Contemporary progress is reported in current scientific journals,e.g.theTransactionsandProceedingsof the Royal Society, and of the Physical Society (London), thePhilosophical Magazine(London), thePhysical Review(New York, 1893 seq.) and in theBritish Association Reports; in theAnnales de chimie et de physique and Journal de physique(Paris); and in thePhysikalische Zeitschrift(Leipzig) and theAnnalen der Physik und Chemie(since 1900:Annalen der Physik) (Leipzig).

(C. E.*)

II.Nature of Light

1.Newton’s Corpuscular Theory.—Until the beginning of the 19th century physicists were divided between two different views concerning the nature of optical phenomena. According to the one, luminous bodies emit extremely small corpuscles which can freely pass through transparent substances and produce the sensation of light by their impact against the retina. Thisemissionorcorpuscular theoryof light was supported by the authority of Isaac Newton,8and, though it has been entirely superseded by its rival, thewave-theory, it remains of considerable historical interest.

2.Explanation of Reflection and Refraction.—Newton supposed the light-corpuscles to be subjected to attractive and repulsive forces exerted at very small distances by the particles of matter. In the interior of a homogeneous body a corpuscle moves in a straight line as it is equally acted on from all sides, but it changes its course at the boundary of two bodies, because, in a thin layer near the surface there is a resultant force in the direction of the normal. In modern language we may say that a corpuscle has at every point a definite potential energy, the value of which is constant throughout the interior of a homogeneous body, and is even equal in all bodies of the same kind, but changes from one substance to another. If, originally, while moving in air, the corpuscles had a definite velocity v0, their velocity v in the interior of any other substance is quite determinate. It is given by the equation ½mv2− ½mv02= A, in which m denotes the mass of a corpuscle, and A the excess of its potential energy in air over that in the substance considered.

A ray of light falling on the surface of separation of two bodies is reflected according to the well-known simple law, if the corpuscles are acted on by a sufficiently large force directed towards the first medium. On the contrary, whenever the field of force near the surface is such that the corpuscles can penetrate into the interior of the second body, the ray is refracted. In this case the law of Snellius can be deduced from the consideration that the projection w of the velocity on the surface of separation is not altered, either in direction or in magnitude. This obviously requires that the plane passing through the incident and the refracted rays be normal to the surface, and that, if α1and α2are the angles of incidence and of refraction, v1and v2the velocities of light in the two media,sin α1/sin α2= w/v1: w/v2= v2/v1.(1)The ratio is constant, because, as has already been observed, v1and v2have definite values.As to the unequal refrangibility of differently coloured light, Newton accounted for it by imagining different kinds of corpuscles. He further carefully examined the phenomenon of total reflection, and described an interesting experiment connected with it. If one of the faces of a glass prism receives on the inside a beam of light of such obliquity that it is totally reflected under ordinary circumstances,a marked change is observed when a second piece of glass is made to approach the reflecting face, so as to be separated from it only by a very thin layer of air. The reflection is then found no longer to be total, part of the light finding its way into the second piece of glass. Newton concluded from this that the corpuscles are attracted by the glass even at a certain small measurable distance.

sin α1/sin α2= w/v1: w/v2= v2/v1.

(1)

The ratio is constant, because, as has already been observed, v1and v2have definite values.

As to the unequal refrangibility of differently coloured light, Newton accounted for it by imagining different kinds of corpuscles. He further carefully examined the phenomenon of total reflection, and described an interesting experiment connected with it. If one of the faces of a glass prism receives on the inside a beam of light of such obliquity that it is totally reflected under ordinary circumstances,a marked change is observed when a second piece of glass is made to approach the reflecting face, so as to be separated from it only by a very thin layer of air. The reflection is then found no longer to be total, part of the light finding its way into the second piece of glass. Newton concluded from this that the corpuscles are attracted by the glass even at a certain small measurable distance.

3.New Hypotheses in the Corpuscular Theory.—The preceding explanation of reflection and refraction is open to a very serious objection. If the particles in a beam of light all moved with the same velocity and were acted on by the same forces, they all ought to follow exactly the same path. In order to understand that part of the incident light is reflected and part of it transmitted, Newton imagined that each corpuscle undergoes certain alternating changes; he assumed that in some of its different “phases” it is more apt to be reflected, and in others more apt to be transmitted. The same idea was applied by him to the phenomena presented by very thin layers. He had observed that a gradual increase of the thickness of a layer produces periodic changes in the intensity of the reflected light, and he very ingeniously explained these by his theory. It is clear that the intensity of the transmitted light will be a minimum if the corpuscles that have traversed the front surface of the layer, having reached that surface while in their phase of easy transmission, have passed to the opposite phase the moment they arrive at the back surface. As to the nature of the alternating phases, Newton (Opticks, 3rd ed., 1721, p. 347) expresses himself as follows:—“Nothing more is requisite for putting the Rays of Light into Fits of easy Reflexion and easy Transmission than that they be small Bodies which by their attractive Powers, or some other Force, stir up Vibrations in what they act upon, which Vibrations being swifter than the Rays, overtake them successively, and agitate them so as by turns to increase and decrease their Velocities, and thereby put them into those Fits.”

4.The Corpuscular Theory and the Wave-Theory compared.—Though Newton introduced the notion of periodic changes, which was to play so prominent a part in the later development of the wave-theory, he rejected this theory in the form in which it had been set forth shortly before by Christiaan Huygens in hisTraité de la lumière(1690), his chief objections being: (1) that the rectilinear propagation had not been satisfactorily accounted for; (2) that the motions of heavenly bodies show no sign of a resistance due to a medium filling all space; and (3) that Huygens had not sufficiently explained the peculiar properties of the rays produced by the double refraction in Iceland spar. In Newton’s days these objections were of much weight.

Yet his own theory had many weaknesses. It explained the propagation in straight lines, but it could assign no cause for the equality of the speed of propagation of all rays. It adapted itself to a large variety of phenomena, even to that of double refraction (Newton says [ibid.]:—“... the unusual Refraction of Iceland Crystal looks very much as if it were perform’d by some kind of attractive virtue lodged in certain Sides both of the Rays, and of the Particles of the Crystal.”), but it could do so only at the price of losing much of its original simplicity.

In the earlier part of the 19th century, the corpuscular theory broke down under the weight of experimental evidence, and it received the final blow when J. B. L. Foucault proved by direct experiment that the velocity of light in water is not greater than that in air, as it should be according to the formula (1), but less than it, as is required by the wave-theory.

5.General Theorems on Rays of Light.—With the aid of suitable assumptions the Newtonian theory can accurately trace the course of a ray of light in any system of isotropic bodies, whether homogeneous or otherwise; the problem being equivalent to that of determining the motion of a material point in a space in which its potential energy is given as a function of the coordinates. The application of the dynamical principles of “least and of varying action” to this latter problem leads to the following important theorems which William Rowan Hamilton made the basis of his exhaustive treatment of systems of rays.9The total energy of a corpuscle is supposed to have a given value, so that, since the potential energy is considered as known at every point, the velocity v is so likewise.

(a) The path along which light travels from a point A to a point B is determined by the condition that for this line the integral ∫v ds, in which ds is an element of the line, be a minimum (provided A and B be not too near each other). Therefore, since v = µv0, if v0is the velocity of lightin vacuoand µ the index of refraction, we have for every variation of the path the points A and B remaining fixed,δ∫µ ds = 0.(2)(b) Let the point A be kept fixed, but let B undergo an infinitely small displacement BB′ (= q) in a direction making an angle θ with the last element of the ray AB. Then, comparing the new ray AB′ with the original one, it follows thatδ∫µ ds = µΒq cos θ,(3)where µΒis the value of µ at the point B.

δ∫µ ds = 0.

(2)

(b) Let the point A be kept fixed, but let B undergo an infinitely small displacement BB′ (= q) in a direction making an angle θ with the last element of the ray AB. Then, comparing the new ray AB′ with the original one, it follows that

δ∫µ ds = µΒq cos θ,

(3)

where µΒis the value of µ at the point B.

6.General Considerations on the Propagation of Waves.—“Waves,”i.e.local disturbances of equilibrium travelling onward with a certain speed, can exist in a large variety of systems. In a theory of these phenomena, the state of things at a definite point may in general be defined by a certain directed or vector quantityP,10which is zero in the state of equilibrium, and may be called the disturbance (for example, the velocity of the air in the case of sound vibrations, or the displacement of the particles of an elastic body from their positions of equilibrium). The componentsPx,Py,Pzof the disturbance in the directions of the axes of coordinates are to be considered as functions of the coordinates x, y, z and the time t, determined by a set of partial differential equations, whose form depends on the nature of the problem considered. If the equations are homogeneous and linear, as they always are for sufficiently small disturbances, the following theorems hold.

(a) Values ofPx,Py,Pz(expressed in terms of x, y, z, t) which satisfy the equations will do so still after multiplication by a common arbitrary constant.(b) Two or more solutions of the equations may be combined into a new solution by addition of the values ofPx, those ofPy, &c.,i.e.by compounding the vectorsP, such as they are in each of the particular solutions.In the application to light, the first proposition means that the phenomena of propagation, reflection, refraction, &c., can be produced in the same way with strong as with weak light. The second proposition contains the principle of the “superposition” of different states, on which the explanation of all phenomena of interference is made to depend.In the simplest cases (monochromatic or homogeneous light) the disturbance is a simple harmonic function of the time (“simple harmonic vibrations”), so that its components can be represented byPx= a1cos (nt + ƒ1),Py= a2cos (nt + ƒ2),Pz= a3cos (nt + ƒ3).The “phases” of these vibrations are determined by the angles nt + ƒ1, &c., or by the times t + ƒ1/n, &c. The “frequency” n is constant throughout the system, while the quantities ƒ1, ƒ2, ƒ3, and perhaps the “amplitudes” a1, a2, a3change from point to point. It may be shown that the end of a straight line representing the vectorP, and drawn from the point considered, in general describes a certain ellipse, which becomes a straight line, if ƒ1= ƒ2= ƒ3. In this latter case, to which the larger part of this article will be confined, we can write in vector notationP=Acos (nt + ƒ),(4)whereAitself is to be regarded as a vector.We have next to consider the way in which the disturbance changes from point to point. The most important case is that of plane waves with constant amplitudeA. Here f is the same at all points of a plane (“wave-front”) of a definite direction, but changes as a linear function as we pass from one such wave-front to the next. The axis of x being drawn at right angles to the wave-fronts, we may write ƒ = ƒ0− kx, where ƒ0and k are constants, so that (4) becomesP = A cos (nt − kx + ƒ0).(5)This expression has the period 2π/n with respect to the time and the perion 2π/k with respect to x, so that the “time of vibration” and the “wave-length” are given by T = 2π/n, λ = 2π/k. Further, it is easily seen that the phase belonging to certain values of x and t is equal to that which corresponds to x + Δx and t + Δt provided Δx = (n/k) Δt. Therefore the phase, or the disturbance itself, may be said to be propagated in the direction normal to the wave-fronts with a velocity (velocity of the waves) v = n/k, which is connected with the time of vibration and the wave-length by the relationλ = vT.(6)In isotropic bodies the propagation can go on in all directions with the same velocity. In anisotropic bodies (crystals), with which the theory of light is largely concerned, the problem is more complicated. As a general rule we can say that, for a given direction of the wave-fronts, the vibrations must have a determinate direction, if the propagation is to take place according to the simple formula given above. It is to be understood that for a given direction of the waves there may be two or even more directions of vibration of the kind, and that in such a case there are as many different velocities, each belonging to one particular direction of vibration.

(a) Values ofPx,Py,Pz(expressed in terms of x, y, z, t) which satisfy the equations will do so still after multiplication by a common arbitrary constant.

(b) Two or more solutions of the equations may be combined into a new solution by addition of the values ofPx, those ofPy, &c.,i.e.by compounding the vectorsP, such as they are in each of the particular solutions.

In the application to light, the first proposition means that the phenomena of propagation, reflection, refraction, &c., can be produced in the same way with strong as with weak light. The second proposition contains the principle of the “superposition” of different states, on which the explanation of all phenomena of interference is made to depend.

In the simplest cases (monochromatic or homogeneous light) the disturbance is a simple harmonic function of the time (“simple harmonic vibrations”), so that its components can be represented by

Px= a1cos (nt + ƒ1),Py= a2cos (nt + ƒ2),Pz= a3cos (nt + ƒ3).

The “phases” of these vibrations are determined by the angles nt + ƒ1, &c., or by the times t + ƒ1/n, &c. The “frequency” n is constant throughout the system, while the quantities ƒ1, ƒ2, ƒ3, and perhaps the “amplitudes” a1, a2, a3change from point to point. It may be shown that the end of a straight line representing the vectorP, and drawn from the point considered, in general describes a certain ellipse, which becomes a straight line, if ƒ1= ƒ2= ƒ3. In this latter case, to which the larger part of this article will be confined, we can write in vector notation

P=Acos (nt + ƒ),

(4)

whereAitself is to be regarded as a vector.

We have next to consider the way in which the disturbance changes from point to point. The most important case is that of plane waves with constant amplitudeA. Here f is the same at all points of a plane (“wave-front”) of a definite direction, but changes as a linear function as we pass from one such wave-front to the next. The axis of x being drawn at right angles to the wave-fronts, we may write ƒ = ƒ0− kx, where ƒ0and k are constants, so that (4) becomes

P = A cos (nt − kx + ƒ0).

(5)

This expression has the period 2π/n with respect to the time and the perion 2π/k with respect to x, so that the “time of vibration” and the “wave-length” are given by T = 2π/n, λ = 2π/k. Further, it is easily seen that the phase belonging to certain values of x and t is equal to that which corresponds to x + Δx and t + Δt provided Δx = (n/k) Δt. Therefore the phase, or the disturbance itself, may be said to be propagated in the direction normal to the wave-fronts with a velocity (velocity of the waves) v = n/k, which is connected with the time of vibration and the wave-length by the relation

λ = vT.

(6)

In isotropic bodies the propagation can go on in all directions with the same velocity. In anisotropic bodies (crystals), with which the theory of light is largely concerned, the problem is more complicated. As a general rule we can say that, for a given direction of the wave-fronts, the vibrations must have a determinate direction, if the propagation is to take place according to the simple formula given above. It is to be understood that for a given direction of the waves there may be two or even more directions of vibration of the kind, and that in such a case there are as many different velocities, each belonging to one particular direction of vibration.

7.Wave-surface.—After having found the values of v for a particular frequency and different directions of the wave-normal, a very instructive graphical representation can be employed.

Let ON be a line in any direction, drawn from a fixed point O, OA a length along this line equal to the velocity v of waves having ON for their normal, or, more generally, OA, OA′, &c., lengths equal to the velocities v, v′, &c., which such waves have according to their direction of vibration, Q, Q′, &c., planes perpendicular to ON through A, A1, &c. Let this construction be repeated for all directions of ON, and let W be the surface that is touched by all the planes Q, Q′, &c. It is clear that if this surface, which is called the “wave-surface,” is known, the velocity of propagation of plane waves of any chosen direction is given by the length of the perpendicular from the centre O on a tangent plane in the given direction. It must be kept in mind that, in general, each tangent plane corresponds to one definite direction of vibration. If this direction is assigned in each point of the wave-surface, the diagram contains all the information which we can desire concerning the propagation of plane waves of the frequency that has been chosen.The plane Q employed in the above construction is the position after unit of time of a wave-front perpendicular to ON and originally passing through the point O. The surface W itself is often considered as the locus of all points that are reached in unit of time by a disturbance starting from O and spreading towards all sides. Admitting the validity of this view, we can determine in a similar way the locus of the points reached in some infinitely short time dt, the wave-surface, as we may say, or the “elementary wave,” corresponding to this time. It is similar to W, all dimensions of the latter surface being multiplied by dt. It may be noticed that in a heterogeneous medium a wave of this kind has the same form as if the properties of matter existing at its centre extended over a finite space.

The plane Q employed in the above construction is the position after unit of time of a wave-front perpendicular to ON and originally passing through the point O. The surface W itself is often considered as the locus of all points that are reached in unit of time by a disturbance starting from O and spreading towards all sides. Admitting the validity of this view, we can determine in a similar way the locus of the points reached in some infinitely short time dt, the wave-surface, as we may say, or the “elementary wave,” corresponding to this time. It is similar to W, all dimensions of the latter surface being multiplied by dt. It may be noticed that in a heterogeneous medium a wave of this kind has the same form as if the properties of matter existing at its centre extended over a finite space.

8.Theory of Huygens.—Huygens was the first to show that the explanation of optical phenomena may be made to depend on the wave-surface, not only in isotropic bodies, in which it has a spherical form, but also in crystals, for one of which (Iceland spar) he deduced the form of the surface from the observed double refraction. In his argument Huygens availed himself of the following principle that is justly named after him: Any point that is reached by a wave of light becomes a new centre of radiation from which the disturbance is propagated towards all sides. On this basis he determined the progress of light-waves by a construction which, under a restriction to be mentioned in § 13, applied to waves of any form and to all kinds of transparent media. Let σ be the surface (wave-front) to which a definite phase of vibration has advanced at a certain time t, dt an infinitely small increment of time, and let an elementary wave corresponding to this interval be described around each point P of σ. Then the envelope σ′ of all these elementary waves is the surface reached by the phase in question at the time t + dt, and by repeating the construction all successive positions of the wave-front can be found.

Huygens also considered the propagation of waves that are laterally limited, by having passed, for example, through an opening in an opaque screen. If, in the first wave-front σ, the disturbance exists only in a certain part bounded by the contour s, we can confine ourselves to the elementary waves around the points of that part, and to a portion of the new wave-front σ′ whose boundary passes through the points where σ′ touches the elementary waves having their centres on s. Taking for granted Huygens’s assumption that a sensible disturbance is only found in those places where the elementary waves are touched by the new wave-front, it may be inferred that the lateral limits of the beam of light are determined by lines, each element of which joins the centre P of an elementary wave with its point of contact P′ with the next wave-front. To lines of this kind, whose course can be made visible by using narrow pencils of light, the name of “rays” is to be given in the wave-theory. The disturbance may be conceived to travel along them with a velocity u = PP′/dt, which is therefore called the “ray-velocity.”The construction shows that, corresponding to each direction of the wave-front (with a determinate direction of vibration), there is a definite direction and a definite velocity of the ray. Both are given by a line drawn from the centre of the wave-surface to its point of contact with a tangent plane of the given direction. It will be convenient to say that this line and the plane are conjugate with each other. The rays of light, curved in non-homogeneous bodies, are always straight lines in homogeneous substances. In an isotropic medium, whether homogeneous or otherwise, they are normal to the wave-fronts, and their velocity is equal to that of the waves.By applying his construction to the reflection and refraction of light, Huygens accounted for these phenomena in isotropic bodies as well as in Iceland spar. It was afterwards shown by Augustin Fresnel that the double refraction in biaxal crystals can be explained in the same way, provided the proper form be assigned to the wave-surface.In any point of a bounding surface the normals to the reflected and refracted waves, whatever be their number, always lie in the plane passing through the normal to the incident waves and that to the surface itself. Moreover, if α1is the angle between these two latter normals, and α2the angle between the normal to the boundary and that to any one of the reflected and refracted waves, and v1, v2the corresponding wave-velocities, the relationsin α1/sin α2= v1/v2(7)is found to hold in all cases. These important theorems may be proved independently of Huygens’s construction by simply observing that, at each point of the surface of separation, there must be a certain connexion between the disturbances existing in the incident, the reflected, and the refracted waves, and that, therefore, the lines of intersection of the surface with the positions of an incident wave-front, succeeding each other at equal intervals of time dt, must coincide with the lines in which the surface is intersected by a similar series of reflected or refracted wave-fronts.In the case of isotropic media, the ratio (7) is constant, so that we are led to the law of Snellius, the index of refraction being given byµ = v1/v2(8)(8)(cf. equation 1).9.General Theorems on Rays, deduced from Huygens’s Construction.—(a) Let A and B be two points arbitrarily chosen in a system of transparent bodies, ds an element of a line drawn from A to B, u the velocity of a ray of light coinciding with ds. Then the integral ∫u−1ds, which represents the time required for a motion along the line with the velocity u, is a minimum for the course actually taken by a ray of light (unless A and B be too far apart). This is the “principle of least time” first formulated by Pierre de Fermat for the case of two isotropic substances. It shows that the course of a ray of light can always be inverted.(b) Rays of light starting in all directions from a point A and travelling onward for a definite length of time, reach a surface σ, whose tangent plane at a point B is conjugate, in the medium surrounding B, with the last element of the ray AB.(c) If all rays issuing from A are concentrated at a point B, the integral ∫u-1ds has the same value for each of them.(d) In case (b) the variation of the integral caused by an infinitely small displacement q of B, the point A remaining fixed, is given by δ∫u−1ds = q cos θ/vB. Here θ is the angle between the displacement q and the normal to the surface σ, in the direction of propagation, vBthe velocity of a plane wave tangent to this surface.In the case of isotropic bodies, for which the relation (8) holds, we recover the theorems concerning the integral ∫µds which we have deduced from the emission theory (§ 5).10.Further General Theorems.—(a) Let V1and V2be two planes in a system of isotropic bodies, let rectangular axes of coordinates be chosen in each of these planes, and let x1, y1be the coordinates of a point A in V1, and x2, y2those of a point B in V2. The integral ∫µds, taken for the ray between A and B, is a function of x1, y1, x2, y2and, if ξ1denotes either x1or y1, and ξ2either x2or y2, we shall have∂2∫µ ds =∂2∫µ ds.∂ξ1∂ξ2∂ξ2∂ξ1On both sides of this equation the first differentiation may be performed by means of the formula (3). The second differentiation admits of a geometrical interpretation, and the formula may finally be employed for proving the following theorem:Let ω1be the solid angle of an infinitely thin pencil of rays issuing from A and intersecting the plane V2in an element σ2at the point B. Similarly, let ω2be the solid angle of a pencil starting from B and falling on the element σ1of the plane V1at the point A. Then, denoting by µ1and µ2the indices of refraction of the matter at the points A and B, by θ1and θ2the sharp angles which the ray AB at its extremities makes with the normals to V1and V2, we haveµ12σ1ω1cos θ1= µ22σ2ω2cos θ2.(b) There is a second theorem that is expressed by exactly the same formula, if we understand by σ1and σ2elements of surface that are related to each other as an object and its optical image—by ω1, ω2the infinitely small openings, at the beginning and the end of its course, of a pencil of rays issuing from a point A of σ1and coming together at the corresponding point B of σ2, and by θ1, θ2the sharp angles which one of the rays makes with the normals to σ1and σ2. The proof may be based upon the first theorem. It suffices toconsider the section σ of the pencil by some intermediate plane, and a bundle of rays starting from the points of σ1and reaching those of σ2after having all passed through a point of that section σ.(c) If in the last theorem the system of bodies is symmetrical around the straight line AB, we can take for σ1and σ2circular planes having AB as axis. Let h1and h2be the radii of these circles,i.e.the linear dimensions of an object and its image, ε1and ε2the infinitely small angles which a ray R going from A to B makes with the axis at these points. Then the above formula gives µ1h1ε1= µ2h2ε2, a relation that was proved, for the particular case µ1= µ2by Huygens and Lagrange. It is still more valuable if one distinguishes by the algebraic sign of h2whether the image is direct or inverted, and by that of ε2whether the ray R on leaving A and on reaching B lies on opposite sides of the axis or on the same side.The above theorems are of much service in the theory of optical instruments and in the general theory of radiation.

The construction shows that, corresponding to each direction of the wave-front (with a determinate direction of vibration), there is a definite direction and a definite velocity of the ray. Both are given by a line drawn from the centre of the wave-surface to its point of contact with a tangent plane of the given direction. It will be convenient to say that this line and the plane are conjugate with each other. The rays of light, curved in non-homogeneous bodies, are always straight lines in homogeneous substances. In an isotropic medium, whether homogeneous or otherwise, they are normal to the wave-fronts, and their velocity is equal to that of the waves.

By applying his construction to the reflection and refraction of light, Huygens accounted for these phenomena in isotropic bodies as well as in Iceland spar. It was afterwards shown by Augustin Fresnel that the double refraction in biaxal crystals can be explained in the same way, provided the proper form be assigned to the wave-surface.

In any point of a bounding surface the normals to the reflected and refracted waves, whatever be their number, always lie in the plane passing through the normal to the incident waves and that to the surface itself. Moreover, if α1is the angle between these two latter normals, and α2the angle between the normal to the boundary and that to any one of the reflected and refracted waves, and v1, v2the corresponding wave-velocities, the relation

sin α1/sin α2= v1/v2

(7)

is found to hold in all cases. These important theorems may be proved independently of Huygens’s construction by simply observing that, at each point of the surface of separation, there must be a certain connexion between the disturbances existing in the incident, the reflected, and the refracted waves, and that, therefore, the lines of intersection of the surface with the positions of an incident wave-front, succeeding each other at equal intervals of time dt, must coincide with the lines in which the surface is intersected by a similar series of reflected or refracted wave-fronts.

In the case of isotropic media, the ratio (7) is constant, so that we are led to the law of Snellius, the index of refraction being given by

µ = v1/v2(8)

(8)

(cf. equation 1).

9.General Theorems on Rays, deduced from Huygens’s Construction.—(a) Let A and B be two points arbitrarily chosen in a system of transparent bodies, ds an element of a line drawn from A to B, u the velocity of a ray of light coinciding with ds. Then the integral ∫u−1ds, which represents the time required for a motion along the line with the velocity u, is a minimum for the course actually taken by a ray of light (unless A and B be too far apart). This is the “principle of least time” first formulated by Pierre de Fermat for the case of two isotropic substances. It shows that the course of a ray of light can always be inverted.

(b) Rays of light starting in all directions from a point A and travelling onward for a definite length of time, reach a surface σ, whose tangent plane at a point B is conjugate, in the medium surrounding B, with the last element of the ray AB.

(d) In case (b) the variation of the integral caused by an infinitely small displacement q of B, the point A remaining fixed, is given by δ∫u−1ds = q cos θ/vB. Here θ is the angle between the displacement q and the normal to the surface σ, in the direction of propagation, vBthe velocity of a plane wave tangent to this surface.

In the case of isotropic bodies, for which the relation (8) holds, we recover the theorems concerning the integral ∫µds which we have deduced from the emission theory (§ 5).

10.Further General Theorems.—(a) Let V1and V2be two planes in a system of isotropic bodies, let rectangular axes of coordinates be chosen in each of these planes, and let x1, y1be the coordinates of a point A in V1, and x2, y2those of a point B in V2. The integral ∫µds, taken for the ray between A and B, is a function of x1, y1, x2, y2and, if ξ1denotes either x1or y1, and ξ2either x2or y2, we shall have

On both sides of this equation the first differentiation may be performed by means of the formula (3). The second differentiation admits of a geometrical interpretation, and the formula may finally be employed for proving the following theorem:

Let ω1be the solid angle of an infinitely thin pencil of rays issuing from A and intersecting the plane V2in an element σ2at the point B. Similarly, let ω2be the solid angle of a pencil starting from B and falling on the element σ1of the plane V1at the point A. Then, denoting by µ1and µ2the indices of refraction of the matter at the points A and B, by θ1and θ2the sharp angles which the ray AB at its extremities makes with the normals to V1and V2, we have

µ12σ1ω1cos θ1= µ22σ2ω2cos θ2.

(b) There is a second theorem that is expressed by exactly the same formula, if we understand by σ1and σ2elements of surface that are related to each other as an object and its optical image—by ω1, ω2the infinitely small openings, at the beginning and the end of its course, of a pencil of rays issuing from a point A of σ1and coming together at the corresponding point B of σ2, and by θ1, θ2the sharp angles which one of the rays makes with the normals to σ1and σ2. The proof may be based upon the first theorem. It suffices toconsider the section σ of the pencil by some intermediate plane, and a bundle of rays starting from the points of σ1and reaching those of σ2after having all passed through a point of that section σ.

(c) If in the last theorem the system of bodies is symmetrical around the straight line AB, we can take for σ1and σ2circular planes having AB as axis. Let h1and h2be the radii of these circles,i.e.the linear dimensions of an object and its image, ε1and ε2the infinitely small angles which a ray R going from A to B makes with the axis at these points. Then the above formula gives µ1h1ε1= µ2h2ε2, a relation that was proved, for the particular case µ1= µ2by Huygens and Lagrange. It is still more valuable if one distinguishes by the algebraic sign of h2whether the image is direct or inverted, and by that of ε2whether the ray R on leaving A and on reaching B lies on opposite sides of the axis or on the same side.

The above theorems are of much service in the theory of optical instruments and in the general theory of radiation.

11.Phenomena of Interference and Diffraction.—The impulses or motions which a luminous body sends forth through the universal medium or aether, were considered by Huygens as being without any regular succession; he neither speaks of vibrations, nor of the physical cause of the colours. The idea that monochromatic light consists of a succession of simple harmonic vibrations like those represented by the equation (5), and that the sensation of colour depends on the frequency, is due to Thomas Young11and Fresnel,12who explained the phenomena of interference on this assumption combined with the principle of super-position. In doing so they were also enabled to determine the wave-length, ranging from 0.000076 cm. at the red end of the spectrum to 0.000039 cm. for the extreme violet and, by means of the formula (6), the number of vibrations per second. Later investigations have shown that the infra-red rays as well as the ultra-violet ones are of the same physical nature as the luminous rays, differing from these only by the greater or smaller length of their waves. The wave-length amounts to 0.006 cm. for the least refrangible infra-red, and is as small as 0.00001 cm. for the extreme ultra-violet.

Another important part of Fresnel’s work is his treatment of diffraction on the basis of Huygens’s principle. If, for example, light falls on a screen with a narrow slit, each point of the slit is regarded as a new centre of vibration, and the intensity at any point behind the screen is found by compounding with each other the disturbances coming from all these points, due account being taken of the phases with which they come together (seeDiffraction;Interference).

12.Results of Later Mathematical Theory.—Though the theory of diffraction developed by Fresnel, and by other physicists who worked on the same lines, shows a most beautiful agreement with observed facts, yet its foundation, Huygens’s principle, cannot, in its original elementary form, be deemed quite satisfactory. The general validity of the results has, however, been confirmed by the researches of those mathematicians (Siméon Denis Poisson, Augustin Louis Cauchy, Sir G. G. Stokes, Gustav Robert Kirchhoff) who investigated the propagation of vibrations in a more rigorous manner. Kirchhoff13showed that the disturbance at any point of the aether inside a closed surface which contains no ponderable matter can be represented as made up of a large number of parts, each of which depends upon the state of things at one point of the surface. This result, the modern form of Huygens’s principle, can be extended to a system of bodies of any kind, the only restriction being that the source of light be not surrounded by the surface. Certain causes capable of producing vibrations can be imagined to be distributed all over this latter, in such a way that the disturbances to which they give rise in the enclosed space are exactly those which are brought about by the real source of light.14Another interesting result that has been verified by experiment is that, whenever rays of light pass through a focus, the phase undergoes a change of half a period. It must be added that the results alluded to in the above, though generally presented in the terms of some particular form of the wave theory, often apply to other forms as well.

13.Rays of Light.—In working out the theory of diffraction it is possible to state exactly in what sense light may be said to travel in straight lines. Behind an openingwhose width is very large in comparison with the wave-lengththe limits between the illuminated and the dark parts of space are approximately determined by rays passing along the borders.

This conclusion can also be arrived at by a mode of reasoning that is independent of the theory of diffraction.15If linear differential equations admit a solution of the form (5) withAconstant, they can also be satisfied by makingAa function of the coordinates, such that, in a wave-front, it changes very little over a distance equal to the wave-length λ, and that it is constant along each line conjugate with the wave-fronts. In cases of this kind the disturbance may truly be said to travel along lines of the said direction, and an observer who is unable to discern lengths of the order of λ, and who uses an opening of much larger dimensions, may very well have the impression of a cylindrical beam with a sharp boundary.A similar result is found for curved waves. If the additional restriction is made that their radii of curvature be very much larger than the wave-length, Huygens’s construction may confidently be employed. The amplitudes all along a ray are determined by, and proportional to, the amplitude at one of its points.

A similar result is found for curved waves. If the additional restriction is made that their radii of curvature be very much larger than the wave-length, Huygens’s construction may confidently be employed. The amplitudes all along a ray are determined by, and proportional to, the amplitude at one of its points.

14.Polarized Light.—As the theorems used in the explanation of interference and diffraction are true for all kinds of vibratory motions, these phenomena can give us no clue to the special kind of vibrations in light-waves. Further information, however, may be drawn from experiments on plane polarized light. The properties of a beam of this kind are completely known when the position of a certain plane passing through the direction of the rays, andinwhich the beam is said to be polarized, is given. “This plane of polarization,” as it is called, coincides with the plane of incidence in those cases where the light has been polarized by reflection on a glass surface under an angle of incidence whose tangent is equal to the index of refraction (Brewster’s law).

The researches of Fresnel and Arago left no doubt as to the direction of the vibrations in polarized light with respect to that of the rays themselves. In isotropic bodies at least, the vibrations are exactly transverse,i.e.perpendicular to the rays, either in the plane of polarization or at right angles to it. The first part of this statement also applies to unpolarized light, as this can always be dissolved into polarized components.

Much experimental work has been done on the production of polarized rays by double refraction and on the reflection of polarized light, either by isotropic or by anisotropic transparent bodies, the object of inquiry being in the latter case to determine the position of the plane of polarization of the reflected rays and their intensity.

In this way a large amount of evidence has been gathered by which it has been possible to test different theories concerning the nature of light and that of the medium through which it is propagated. A common feature of nearly all these theories is that the aether is supposed to exist not only in spaces void of matter, but also in the interior of ponderable bodies.

15.Fresnel’s Theory.—Fresnel and his immediate successors assimilated the aether to an elastic solid, so that the velocity of propagation of transverse vibrations could be determined by the formula v = √(K/ρ), where K denotes the modulus of rigidity and ρ the density. According to this equation the different properties of various isotropic transparent bodies may arise from different values of K, of ρ, or of both. It has, however, been found that if both K and ρ are supposed to change from one substance to another, it is impossible to obtain the right reflection formulae. Assuming the constancy of K Fresnel was led to equations which agreed with the observed properties of the reflected light, if he made the further assumption (to be mentioned in what follows as “Fresnel’s assumption”) that the vibrations of plane polarized light are perpendicular to the plane of polarization.

Let the indices p and n relate to the two principal cases in which the incident (and, consequently, the reflected) light is polarized in the plane of incidence, or normally to it, and let positive directions h and h′ be chosen for the disturbance (at the surface itself) in the incident and for that in the reflected beam, in such a manner that, by a common rotation, h and the incident ray prolonged may be made to coincide with h′ and the reflected ray. Then, if α1and α2are the angles of incidence and refraction, Fresnel shows that, in order to get the reflected disturbance, the incident one must be multiplied byαp= −sin (α1− α2) / sin (α1+ α2)(9)in the first, and byαn= tan (α1− α2) / tan (α1+ α2)(10)in the second principal case.

αp= −sin (α1− α2) / sin (α1+ α2)

(9)

in the first, and by

αn= tan (α1− α2) / tan (α1+ α2)

(10)

in the second principal case.

As to double refraction, Fresnel made it depend on the unequal elasticity of the aether in different directions. He came to the conclusion that, for a given direction of the waves, there are two possible directions of vibration (§ 6), lying in the wave-front, at right angles to each other, and he determined the form of the wave-surface, both in uniaxal and in biaxal crystals.

Though objections may be urged against the dynamic part of Fresnel’s theory, he admirably succeeded in adapting it to the facts.

16. Electromagnetic Theory.—We here leave the historical order and pass on to Maxwell’s theory of light.

James Clerk Maxwell, who had set himself the task of mathematically working out Michael Faraday’s views, and who, both by doing so and by introducing many new ideas of his own, became the founder of the modern science of electricity,16recognized that, at every point of an electromagnetic field, the state of things can be defined by two vector quantities, the “electric force”Eand the “magnetic force”H, the former of which is the force acting on unit of electricity and the latter that which acts on a magnetic pole of unit strength. In a non-conductor (dielectric) the forceEproduces a state that may be described as a displacement of electricity from its position of equilibrium. This state is represented by a vectorD(“dielectric displacement”) whose magnitude is measured by the quantity of electricity reckoned per unit area which has traversed an element of surface perpendicular toDitself. Similarly, there is a vector quantityB(the “magnetic induction”) intimately connected with the magnetic forceH. Changes of the dielectric displacement constitute an electric current measured by the rate of change ofD, and represented in vector notation byC=Ḋ(11)Periodic changes ofDandBmay be called “electric” and “magnetic vibrations.” Properly choosing the units, the axes of coordinates (in the first proposition also the positive direction of s and n), and denoting components of vectors by suitable indices, we can express in the following way the fundamental propositions of the theory.(a) Let s be a closed line, σ a surface bounded by it, n the normal to σ. Then, for all bodies,∫Hsds =1∫Cndσ,∫Esds = −1d∫Bndσ,ccdtwhere the constant c means the ratio between the electro-magnet and the electrostatic unit of electricity.From these equations we can deduce:(α) For the interior of a body, the equations∂Hz−∂Hy=1Cx,∂Hx−∂Hz=1Cy,∂Hy−∂Hx=1Cz∂y∂zc∂z∂xc∂x∂yc(12)∂Ez−∂Ey= −1∂Bz,∂Ex−∂Ez= −1∂By,∂Ey−∂Ex= −1∂Bz;∂y∂zc∂t∂z∂xc∂t∂x∂yc∂t(13)(ß) For a surface of separation, the continuity of the tangential components ofEandH;(γ) The solenoidal distribution ofCandB, and in a dielectric that ofD. A solenoidal distribution of a vector is one corresponding to that of the velocity in an incompressible fluid. It involves the continuity, at a surface, of the normal component of the vector.(b) The relation between the electric force and the dielectric displacement is expressed byDx= ε1Ex,Dy= ε2Ey,Dz= ε3Ez,(14)the constants ε1, ε2, ε3(dielectric constants) depending on the properties of the body considered. In an isotropic medium they have a common value ε, which is equal to unity for the free aether, so that for this mediumD=E.(c) There is a relation similar to (14) between the magnetic force and the magnetic induction. For the aether, however, and for all ponderable bodies with which this article is concerned, we may writeB=H.It follows from these principles that, in an isotropicdielectric, transverse electric vibrations can be propagated with a velocityv = c / √ε.(15)Indeed, all conditions are satisfied if we putDx= 0,Dy= a cos n (t − xv−1+ l),Dz= 0,Hx= 0,Hy= 0,Hz= avc−1cos n (t − xv−1+ l)(16)For the free aether the velocity has the value c. Now it had been found that the ratio c between the two units of electricity agrees within the limits of experimental errors with the numerical value of the velocity of light in aether. (The mean result of the most exact determinations17of c is 3,001·1010cm./sec., the largest deviations being about 0,008·1010; and Cornu18gives 3,001·1010± 0,003·1010as the most probable value of the velocity of light.) By this Maxwell was led to suppose that light consists of transverse electromagnetic disturbances. On this assumption, the equations (16) represent a beam of plane polarized light. They show that, in such a beam, there are at the same time electric and magnetic vibrations, both transverse, and at right angles to each other.It must be added that the electromagnetic field is the seat of two kinds of energy distinguished by the names of electric and magnetic energy, and that, according to a beautiful theorem due to J. H. Poynting,19the energy may be conceived to flow in a direction perpendicular both to the electric and to the magnetic force. The amounts per unit of volume of the electric and the magnetic energy are given by the expressions½ (ExDx+EyDy+EzDz),(17)and½ (HxBx+HyBy+HzBz) = ½H2,(18)whose mean values for a full period are equal in every beam of light.The formula (15) shows that the index of refraction of a body is given by √ε, a result that has been verified by Ludwig Boltzmann’s measurements20of the dielectric constants of gases. Thus Maxwell’s theory can assign the true cause of the different optical properties of various transparent bodies. It also leads to the reflection formulae (9) and (10), provided the electric vibrations of polarized light be supposed to be perpendicular to the plane of polarization, which implies that the magnetic vibrations are parallel to that plane.Following the same assumption Maxwell deduced the laws of double refraction, which he ascribes to the unequality of ε1, ε2, ε3. His results agree with those of Fresnel and the theory has been confirmed by Boltzmann,21who measured the three coefficients in the case of crystallized sulphur, and compared them with the principal indices of refraction. Subsequently the problem of crystalline reflection has been completely solved and it has been shown that, in a crystal, Poynting’s flow of energy has the direction of the rays as determined by Huygens’s construction.Two further verifications must here be mentioned. In the first place, though we shall speak almost exclusively of the propagation of light in transparent dielectrics, a few words may be said about the optical properties of conductors. The simplest assumption concerning the electric currentCin a metallic body is expressed by the equationC= σE, where σ is the coefficient of conductivity. Combining this with his other formulae (we may say with (12) and (13)), Maxwell found that there must be an absorption of light, a result that can be readily understood since the motion of electricity in a conductor gives rise to a development of heat. But, though Maxwell accounted in this way for the fundamental fact that metals are opaque bodies, there remained a wide divergence between the values of the coefficient of absorption as directly measured and as calculated from the electrical conductivity; but in 1903 it was shown by E. Hagen and H. Rubens22that the agreement is very satisfactory in the case of the extreme infra-red rays.In the second place, the electromagnetic theory requires that a surface struck by a beam of light shall experience a certain pressure. If the beam falls normally on a plane disk, the pressure is normal too; its total amount is given by c−1(i1+ i2− i3), if i1, i2and i3are the quantities of energy that are carried forward per unit of time by the incident, the reflected, and the transmitted light. This result has been quantitatively verified by E. F. Nicholls and G. F. Hull.23Maxwell’s predictions have been splendidly confirmed by the experiments of Heinrich Hertz24and others on electromagnetic waves; by diminishing the length of these to the utmost, some physicists have been able to reproduce with them all phenomena of reflection, refraction (single and double), interference, and polarization.25A table of the wave-lengths observed in the aether now hasto contain, besides the numbers given in § 11, the lengths of the waves produced by electromagnetic apparatus and extending from the long waves used in wireless telegraphy down to about 0.6 cm.

C=Ḋ

(11)

Periodic changes ofDandBmay be called “electric” and “magnetic vibrations.” Properly choosing the units, the axes of coordinates (in the first proposition also the positive direction of s and n), and denoting components of vectors by suitable indices, we can express in the following way the fundamental propositions of the theory.

(a) Let s be a closed line, σ a surface bounded by it, n the normal to σ. Then, for all bodies,

where the constant c means the ratio between the electro-magnet and the electrostatic unit of electricity.

From these equations we can deduce:

(α) For the interior of a body, the equations

(12)

(13)

(ß) For a surface of separation, the continuity of the tangential components ofEandH;

(γ) The solenoidal distribution ofCandB, and in a dielectric that ofD. A solenoidal distribution of a vector is one corresponding to that of the velocity in an incompressible fluid. It involves the continuity, at a surface, of the normal component of the vector.

(b) The relation between the electric force and the dielectric displacement is expressed by

Dx= ε1Ex,Dy= ε2Ey,Dz= ε3Ez,

(14)

the constants ε1, ε2, ε3(dielectric constants) depending on the properties of the body considered. In an isotropic medium they have a common value ε, which is equal to unity for the free aether, so that for this mediumD=E.

(c) There is a relation similar to (14) between the magnetic force and the magnetic induction. For the aether, however, and for all ponderable bodies with which this article is concerned, we may writeB=H.

It follows from these principles that, in an isotropicdielectric, transverse electric vibrations can be propagated with a velocity

v = c / √ε.

(15)

Indeed, all conditions are satisfied if we put

(16)

For the free aether the velocity has the value c. Now it had been found that the ratio c between the two units of electricity agrees within the limits of experimental errors with the numerical value of the velocity of light in aether. (The mean result of the most exact determinations17of c is 3,001·1010cm./sec., the largest deviations being about 0,008·1010; and Cornu18gives 3,001·1010± 0,003·1010as the most probable value of the velocity of light.) By this Maxwell was led to suppose that light consists of transverse electromagnetic disturbances. On this assumption, the equations (16) represent a beam of plane polarized light. They show that, in such a beam, there are at the same time electric and magnetic vibrations, both transverse, and at right angles to each other.

It must be added that the electromagnetic field is the seat of two kinds of energy distinguished by the names of electric and magnetic energy, and that, according to a beautiful theorem due to J. H. Poynting,19the energy may be conceived to flow in a direction perpendicular both to the electric and to the magnetic force. The amounts per unit of volume of the electric and the magnetic energy are given by the expressions

½ (ExDx+EyDy+EzDz),

(17)

and

½ (HxBx+HyBy+HzBz) = ½H2,

(18)

whose mean values for a full period are equal in every beam of light.

The formula (15) shows that the index of refraction of a body is given by √ε, a result that has been verified by Ludwig Boltzmann’s measurements20of the dielectric constants of gases. Thus Maxwell’s theory can assign the true cause of the different optical properties of various transparent bodies. It also leads to the reflection formulae (9) and (10), provided the electric vibrations of polarized light be supposed to be perpendicular to the plane of polarization, which implies that the magnetic vibrations are parallel to that plane.

Following the same assumption Maxwell deduced the laws of double refraction, which he ascribes to the unequality of ε1, ε2, ε3. His results agree with those of Fresnel and the theory has been confirmed by Boltzmann,21who measured the three coefficients in the case of crystallized sulphur, and compared them with the principal indices of refraction. Subsequently the problem of crystalline reflection has been completely solved and it has been shown that, in a crystal, Poynting’s flow of energy has the direction of the rays as determined by Huygens’s construction.

Two further verifications must here be mentioned. In the first place, though we shall speak almost exclusively of the propagation of light in transparent dielectrics, a few words may be said about the optical properties of conductors. The simplest assumption concerning the electric currentCin a metallic body is expressed by the equationC= σE, where σ is the coefficient of conductivity. Combining this with his other formulae (we may say with (12) and (13)), Maxwell found that there must be an absorption of light, a result that can be readily understood since the motion of electricity in a conductor gives rise to a development of heat. But, though Maxwell accounted in this way for the fundamental fact that metals are opaque bodies, there remained a wide divergence between the values of the coefficient of absorption as directly measured and as calculated from the electrical conductivity; but in 1903 it was shown by E. Hagen and H. Rubens22that the agreement is very satisfactory in the case of the extreme infra-red rays.

In the second place, the electromagnetic theory requires that a surface struck by a beam of light shall experience a certain pressure. If the beam falls normally on a plane disk, the pressure is normal too; its total amount is given by c−1(i1+ i2− i3), if i1, i2and i3are the quantities of energy that are carried forward per unit of time by the incident, the reflected, and the transmitted light. This result has been quantitatively verified by E. F. Nicholls and G. F. Hull.23

Maxwell’s predictions have been splendidly confirmed by the experiments of Heinrich Hertz24and others on electromagnetic waves; by diminishing the length of these to the utmost, some physicists have been able to reproduce with them all phenomena of reflection, refraction (single and double), interference, and polarization.25A table of the wave-lengths observed in the aether now hasto contain, besides the numbers given in § 11, the lengths of the waves produced by electromagnetic apparatus and extending from the long waves used in wireless telegraphy down to about 0.6 cm.

17.Mechanical Models of the Electromagnetic Medium.—From the results already enumerated, a clear idea can be formed of the difficulties which were encountered in the older form of the wave-theory. Whereas, in Maxwell’s theory, longitudinal vibrations are excludedab initioby the solenoidal distribution of the electric current, the elastic-solid theory had to take them into account, unless, as was often done, one made them disappear by supposing them to have a very great velocity of propagation, so that the aether was considered to be practically incompressible. Even on this assumption, however, much in Fresnel’s theory remained questionable. Thus George Green,26who was the first to apply the theory of elasticity in an unobjectionable manner, arrived on Fresnel’s assumption at a formula for the reflection coefficient Ansensibly differing from (10).

In the theory of double refraction the difficulties are no less serious. As a general rule there are in an anisotropic elastic solid three possible directions of vibration (§ 6), at right angles to each other, for a given direction of the waves, but none of these lies in the wave-front. In order to make two of them do so and to find Fresnel’s form for the wave-surface, new hypotheses are required. On Fresnel’s assumption it is even necessary, as was observed by Green, to suppose that in the absence of all vibrations there is already a certain state of pressure in the medium.

Back to Index Next