Wavertree

WAVE THEORY OF LIGHT

§ 1. A GENERAL statement of the principles of the undulatory theory, with elementary explanations, has already been given under Light, and in the article on Ether the arguments which point to the existence of an all-pervading medium, susceptible in its various parts of an alternating change of state, have been traced by a master hand; but the subject is of such great importance, and is so intimately involved in recent optical investigation and discovery, that a more detailed exposition of the theory, with application to the leading phenomena, was reserved for a special article. That the subject is one of difficulty may be at once admitted. Even in the theory of sound, as conveyed by aerial vibrations, where we are well acquainted with the nature and properties of the vehicle, the fundamental conceptions are not very easy to grasp, and their development makes heavy demands upon our mathematical resources. That the situation is not improved when the medium is hypothetical will be easily understood. For, although the evidence is overwhelming in favour of the conclusion that light is propagated as a vibration, we are almost entirely in the dark as to what it is that vibrates and the manner of vibration. This ignorance entails an appearance of vagueness even in those parts of the subject the treatment of which would not really be modified by the acquisition of a more precise knowledge, e.g., the theory of the colours of thin plates, and of the resolving power of optical instruments. But in other parts of the subject, such as the explanation of the laws of double refraction and of the intensity of light reflected at the surface of a transparent medium, the vagueness is not merely one of language; and if we wish to reach definite results by the a prion road we must admit a hypothetical element, for which little justification can be given. The distinction here indicated should be borne clearly in mind. Many optical phenomena must necessarily agree with any kind of wave theory that can be proposed; others may agree or disagree with a particular form of it. In the latter case we may regard the special form as disproved, but the undulatory theory in the proper wider sense remains untouched.

Of such special forms of the wave theory the most famous is that which assimilates light to the transverse vibrations of an elastic solid. Transverse they must be in order to give room for the phenomena of polarization. This theory is a great help to the imagination, and allows of the deduction of definite results which are at any rate mechanically possible. An isotropic solid has in general two elastic properties—one relating to the recovery from an alteration of volume, and the other to the recovery from a state of shear, in which the strata are caused to slide over one another. It has been shown by Green that it would be necessary to suppose the luminiferous medium to be incompressible, and thus the only admissible differences between one isotropic medium and another are those of rigidity and of density. Between these we are in the first instance free to choose. The slower propagation of light in glass than in air may be equally well explained by supposing the rigidity the same in both cases while the density is greater in glass, or by supposing that the density is the same in both cases while the rigidity is greater in air. Indeed there is nothing, so far, to exclude a more complicated condition of things, in which both the density and rigidity vary in passing from one medium to another, subject to the one condition only of making the ratio of velocities of propagation equal to the known refractive index between the media.

When we come to apply this theory to investigate the intensity of light reflected from (say) a glass surface, and to the diffraction of light by very small particles (as in the sky), we find that a reasonable agreement with the facts can be brought about only upon the supposition that the rigidity is the same (approximately, at any rate) in various media, and that the density alone varies. At the same time we have to suppose that the vibration is perpendicular to the plane of polarization.

Up to this point the accordance may be regarded as fairly satisfactory; but, when we extend the investigation to crystalline media in the hope of explaining the observed laws of double refraction, we find that the suppositions which would suit best here are inconsistent with the conclusions we have already arrived at. In the first place, and so long as we hold strictly to the analogy of an elastic solid, we can only explain double refraction as depending upon anisotropic rigidity, and this can hardly be reconciled with the view that the rigidity is the same in different isotropic media. And if we pass over this difficulty, and inquire what kind of double refraction a crystalline solid would admit of, we find no such correspondence with observation as would lead us to think that we are upon the right track. The theory of anisotropic solids, with its twenty-one elastic constants, seems to be too wide for optical double refraction, which is of a much simpler character.

For these and other reasons, especially the awkwardness with which it lends itself to the explanation of dispersion, the elastic solid theory, valuable as a piece of purely dynamical reasoning, and probably not without mathematical analogy to the truth, can in optics be regarded only as an illustration.

In recent years a theory has been received with much favour in which light is regarded as an electromagnetic phenomenon. The dielectric medium is conceived to be subject to a rapidly periodic “electric displacement,” the variations of which have the magnetic properties of an electric current. On the basis of purely electrical observations Maxwell calculated the velocity of propagation of such disturbances, and obtained a value not certainly distinguishable from the velocity of light. Such an agreement is very striking; and a further deduction from the theory, that the specific inductive capacity of a transparent medium is equal to the square of the refractive index, is supported to some extent by observation. The foundations of the electrical theory are not as yet quite cleared of more or less arbitrary hypothesis; but, when it becomes certain that a dielectric medium is susceptible of vibrations propagated with the velocity of light, there will be no hesitation in accepting the identity of such vibrations with those to which optical phenomena are due. In the meantime, and apart altogether from the question of its probable truth, the electromagnetic theory is very instructive, in showing us how careful we must be to avoid limiting our ideas too much as to the nature of the luminous vibrations.

§ 2. Plane Waves of Simple Type.

Whatever may be the character of the medium and of its vibration, the analytical expression for an infinite train of plane waves is Acos j y(Vi-α) + α J (1), in which λ represents the wave-length, and V the corresponding velocity of propagation. The coefficient A is called the amplitude, and its nature depends upon the medium and must therefore here be left an open question. The phase of the wave at a given time and place is represented by a. The expression retains the same value whatever integral number of wave-lengths be added to or subtracted from x. It is also periodic with respect to t, and the period is τ = λ ∕y (2)<

In experimenting upon sound we are able to determine independently τ, λ, and V; but on account of its smallness the periodic time of luminous vibrations eludes altogether our means of observation, and is only known indirectly from λ and V by means of (2).

There is nothing arbitrary in the use of a circular function to represent the waves. As a general rule this is the only kind of wave which can be propagated without a change of form; and, even in the exceptional cases where the velocity is independent of wavelength, no generality is really lost by this procedure, because in accordance with Fourier’s theorem any kind of periodic wave may be regarded as compounded of a series of such as (1), with wavelengths in harmonical progression.

A well-known characteristic of waves of type (1) is that any number of trains of various amplitudes and phases, but of the same wave-length, are equivalent to a single train of the same type. Thus SAcos ^(Vi-αr) + α

= SA cos a. cos -f ( Vt - x) - SA sin «. sin— (Vt-x) A λ

= Pcos j y^(VZ-≈) + 0 I ..... (3), where P 2 = (SAcosa) 2 + S(Asina) 2 .... (4),

S(Asina)

tan φ = -½ ( 5 ).

S(Acosa)

An important particular case is that of two component trains only. Acos ~(Υt~x') + a y +A'cos ^(Vi-iB) + α' ∣

= Pcos j ~(Vt~x) + φ , where P 2 =≈A 2 ÷A' 2 + 2AA'cos(a-a') .... (6).

The composition of vibrations of the same period is precisely analogous, as was pointed out by Fresnel, to the composition of forces, or indeed of any other two-dimensional vector quantities. The magnitude of the force corresponds to the amplitude of the vibration, and the inclination of the force corresponds to the phase. Λ group of forces, of equal intensity, represented by lines drawn from the centre to the angular points of a regular polygon, constitute a system in equilibrium. Consequently, a system of vibrations of equal amplitude and of phases symmetrically distributed round the period has a zero resultant.

According to the phase-relation, determined by (α-α'), the amplitude of the resultant may vary from (A - A') to (A +A'). If A' and A are equal, the minimum resultant is zero, showing that two equal trains of waves may neutralize one another. This happens when the phases are opposite, or differ by half a (complete) period, and the effect is usually spoken of as the interference of light. From a purely dynamical point of view the word is not very appropriate, the vibrations being simply superposed with as little interference as can be imagined.

§ 3. Intensity.

The intensity of light of given wave-length must depend upon the amplitude, but the precise nature of the relation is not at once apparent. We are not able to appreciate by simple inspection the relative intensities of two unequal lights; and, when we say, for example, that one candle is twice as bright as another, we mean that two of the latter burning independently would give us the same light as one of the former. This may be regarded as the definition; and then experiment may be appealed to to prove that the intensity of light from a given source varies inversely as the square of the distance. But our conviction of the truth of the law is perhaps founded quite as much upon the idea that something not liable to loss is radiated outwards, and is distributed in succession over the surfaces of spheres concentric with the source, whose areas are as the squares of the radii. The something can only be energy; and thus we are led to regard the rate at which energy is propagated across a given area parallel to the waves as the measure of intensity; and this is proportional, not to the first power, but to the square of the amplitude.

Practical photometry is usually founded upon the law of inverse squares (Light, vol. xiv. p. 583); and it should be remembered that the method involves essentially the use of a diffusing screen, the illumination of which, seen in a certain direction, is assumed to be independent of the precise direction in which the light falls upon it; for the distance of a candle, for example, cannot be altered without introducing at the same time a change in the apparent magnitude, and therefore in the incidence of some part at any rate of the light.

With this objection is connected another which is often of greater importance, the necessary enfeeblement of the light by the process of diffusion. Aud, if to maintain the brilliancy we substitute regular reflectors for diffusing screens, the method breaks down altogether by the apparent illumination becoming independent of the distance of the source of light.

The use of a revolving disk with transparent and opaque sectors in order to control the brightness, as proposed by Fox Talbot, may often be recommended in scientific photometry, when a great loss of light is inadmissible. The law that, when the frequency of intermittence is sufficient to give a steady appearance, the brightness is proportional to the angular magnitude of the open sectors appears to be well established.

§ 4. Resultant of a Large Humber of Vibrations of Arbitrary Phase.

We have seen that the resultant of two vibrations of equal amplitude is wholly dependent upon their phase-relation, and it is of interest to inquire what we are to expect from the composition of a large number (n) of equal vibrations of amplitude unity, and of arbitrary phases. The intensity of the resultant will of course depend upon the precise manner in which the phases are distributed, and may vary from n 2 to zero. But is there a definite intensity which becomes more and more probable as n is increased without limit?

The nature of the question here raised is well illustrated by the special case in which the possible phases are restricted to two opposite phases. We may then conveniently discard the idea of phase, and regard the amplitudes as at random positive or negative. If all the signs are the same, the intensity is n 2 ; if, on the other hand, there are as many positive as negative, the result is zero. But, although the intensity may range from 0 to n 2 , the smaller values are much more probable than the greater.

The simplest part of the problem relates to what is called in the theory of probabilities the “expectation” of intensity, that is, the mean intensity to be expected after a great number of trials, in each of which the phases are taken at random. The chance that all the vibrations are positive is 2 -n , and thus the expectation of intensity corresponding to this contingency is 2 -n . n 2 . In like manner the expectation corresponding to the number of positive vibrations being (n-1) is 2 -".n.(π-2) 2 , and so on. The whole expectation of intensity is thus j l.B≈+n.(a-2r+⅛L≤∖ n -4)≈ ÷-⅜2⅞- l ι ,t - 6 > 1 ÷ · · · J · · · (»· Now the sum of the (n + l) terms of this series is simply n, as may be proved by comparison of coefficients of x 2 in the equivalent forms

(e≈+e-≈) n =2 n (l + ⅜x 2 + . . .) n =β"≈+jιe(,, - 2 ) ar +-^- 2 1 ^(,, - 4 ) x + . . .

The expectation of intensity is therefore n, and this whether n be great or small.

The same conclusion holds good when the phases are unrestricted. From (4), § 2, if A = 1,

P 2 = 7i∙+22 cos (<x 2 — α 1 ) (2), where under the sign of summation are to be included the cosines of the ½n(n - 1) differences of phase. When the phases are arbitrary, this sum is as likely to be positive as negative, and thus the mean value of P 2 is n.

The reader must be on his guard here against a fallacy which has misled some high authorities. We have not proved that when n is large there is any tendency for a single combination to give the intensity equal to n, but the quite different proposition that in a large number of trials, in each of which the phases are rearranged arbitrarily, the mean intensity will tend more and more to the value n. It is true that even in a single combination there is no reason why any of the cosines in (2) should be positive rather than negative, and from this we may infer that when n is increased the sum of the terms tends to vanish in comparison with the number of terms. But, the number of terms being of the order n 2 , we can infer nothing as to the value of the sum of the series in comparison with n.

Indeed it is not true that the intensity in a single combination approximates to n, when n is large. It can be proved that the probability of a resultant intermediate in amplitude between r and r + dr is - e~ r 'l n rdr ....... (3). n

The probability of an amplitude less than r is thus - ∕' r e- r2 l n rdr≈l-e~ r2 ' n (4).

n/o or, which is the same thing, the probability of an amplitude greater than r is e - r2/n . . . . . . . . (5).

The accompanying table gives the probabilities of intensities less than the fractious of n named in the first column. For example, the probability of intensity less than n is ∙6321.

It will be seen that, however great n may be, there is a fair chance of considerable relative fluctuations of intensity in consecutive combinations. [table]

The mean intensity, expressed by - Γ*e~ r2 l n .r i .rdr, udo is, as we have already seen, equal to τι.

It is with this mean intensity only that we are concerned in ordinary photometry. A source of light, such as a candle or even a soda flame, may be regarded as composed of a very large number of luminous centres disposed throughout a very sensible space; and, even though it be true that the intensity at a particular point of a screen illuminated by it and at a particular moment of time is a matter of chance, further processes of averaging must be gone through before anything is arrived at of which our senses could ordinarily take cognizance. In the smallest interval of time during which the eye could be impressed, there would be opportunity for any number of rearrangements of phase, due either to motions of the particles or to irregularities in their modes of vibration. And even if we supposed that each luminous centre was fixed, and emitted perfectly regular vibrations, the manner of composition and consequent intensity would vary rapidly from point to point of the screen, and in ordinary cases the mean illumination over the smallest appreciable area would correspond to a thorough averaging of the phase-relationships. In this way the idea of the intensity of a luminous source, independently of any questions of phase, is seen to be justified, and we may properly say that two candles are twice as bright as one.

§ 5. Propagation of Waves in General.

It has been shown under Optics that a system of rays, however many reflexions or refractions they may have undergone, are always normal to a certain surface, or rather system of surfaces. From our present point of view these surfaces are to be regarded as wave-surfaces, that is, surfaces of constant phase. It is evident that, so long as the radius of curvature is very large in comparison with λ, each small part of a wave-surface propagates itself just as an infinite plane wave coincident with the tangent plane would do. If we start at time t with a given surface, the corresponding wavesurface at time t+dt is to be found by prolonging every normal by the length V dt, where V denotes the velocity of propagation at the place in question. If the medium be uniform, so that V is constant, the new surface is parallel to the old one, and this property is retained however many short intervals of time be considered in succession. A wave-surface thus propagates itself normally, and the corresponding parts of successive surfaces are those whieh lie upon the same normal. In this sense the normal may be regarded as a ray, but the idea must not be pushed to streams of light limited to pass through small apertures. The manner in which the phase is determined by the length of the ray, and the conditions under which energy may be regarded as travelling along a ray, will be better treated under the head of Huygens’s principle, and the theory of shadows (§ 10).

From the law of propagation, according to which the wavesurfaces are always as far advanced as possible, it follows that the course of a ray is that for which the time, represented by ∫ V -1 ds, is a minimum. This is Fermat’s principle of least time. Since the refractive index (μ) varies as V -1 , we may take ∫μ ds as the measure of the retardation between one wave surface and another; and it is the same along whichever ray it may be measured.

The principle that ∫μds is a minimum along a ray lends itself readily to the investigation of optical laws. As an example, we will consider the very important theory of magnifying power. Let A 0 ,B 0 be two points upon a wave-surface before the light enters the object-glass of a telescope, A, B the corresponding points upon a wave-surface after emergence from the eye-piece, both surfaces being plane. The value of ∫μds is the same along the ray A 0 A as along B 0 B; and, if from any cause B 0 be slightly retarded relatively to A 0 , then B will be retarded to the same amount relatively to A. Suppose now that the retardation in question is due to a small rotation (θ ) of the wave-surface A 0 B 0 about an axis in its own plane perpendicular to AB. The retardation of B 0 relatively to A 0 is then A 0 B 0 . θ ; and in like manner, if φ be the corresponding rotation of AB, the retardation is AB. φ . Since these retardations are the same, we have Φ _ A(∣Bq θ AB ’ or the magnifying power is equal to the ratio of the widths of the stream of light before and after passing the telescope.

The magnifying power is not necessarily the same in all directions. Consider the case of a prism arranged as for spectrum work. Passage through the prism does not alter the vertical width of the stream of light; hence there is no magnifying power in this direction. What happens in a horizontal direction depends upon circumstances. A single prism in the position of minimum deviation does not alter the horizontal width of the beam. The same is true of a sequence of any number of prisms each in the position of minimum deviation, or of the combination called by Thollon a couple, when the deviation is the least that can be obtained by rotating the couple as a rigid system, although a further diminution might be arrived at by violating this tie. In all these cases there is neither horizontal nor vertical magnification, and the instrument behaves as a telescope of power unity. If, however, a prism be so placed that the angle of emergence differs from the angle of incidence, the horizontal width of the beam undergoes a change. If the emergence be nearly grazing, there will be a high magnifying power in the horizontal direction; and, whatever may be the character of the system of prisms, the horizontal magnifying power is represented by the ratio of widths. Brewster suggested that, by combining two prisms with refracting edges at right angles, it would be possible to secure equal magnifying power in the two directions, and thus to imitate the action of an ordinary telescope.

The theory of magnifying power is intimately connected with that of apparent brightness. By the use of a telescope in regarding a bright body, such, for example, as the moon, there is a concentration of light upon the pupil in proportion to the ratio of the area of the object-glass to that of the pupil. But the apparent brightness remains unaltered, the apparent superficial magnitude of the object being changed in precisely the same proportion, in accordance with the law just established.

These fundamental propositions were proved a long while since by Cotes and Smith; and a complete exposition of them, from the point of view of geometrical optics, is to be found in Smith’s treatise.

§ 6. Waves Approximately Plane or Spherical.

A plane wave of course remains plane after reflexion from a truly plane surface; but any irregularities in the surface impress themselves upon the wave. In the simplest case, that of perpendicular incidence, the irregularities are doubled, any depressed portion of the surface giving rise to a retardation in the wave front of twice its own amount. It is assumed that the lateral dimensions of the depressed or elevated parts are large multiples of the wave-length; otherwise the assimilation of the various parts to plane waves is not legitimate.

In like manner, if a plane wave passes perpendicularly through a parallel plate of refracting material, a small elevation t at any part of one of the surfaces introduces a retardation (μ - 1) t in the corresponding part of the wave-surface. An error in a glass surface is thus of only one-quarter of the importance of an equal error in a reflecting surface. Further, if a plate, otherwise true, be distorted by bending, the errors introduced at the two surfaces are approximately opposite, and neutralize one another.

In practical applications it is of importance to recognize the effects of a small departure of the wave-surface from its ideal plane or spherical form. Let the surface be referred to a system of rectangular coordinates, the axis of z being normal at the centre of the section of the beam, and the origin being the point of contact of the tangent plane. If, as happens in many cases, the surface be one of symmetry round OZ, the equation of the surface may be represented approximately by

≈ = r 2 ∕2p +Ar 4 + (1), in which ρ is the radius of curvature, or focal length, and r 2 =x 2 + y 2 . If the surface be truly spherical, A = 1∕8ρ 3 , and any deviation of A from this value indicates ordinary symmetrical spherical aberration.

If, however, the surface be not symmetrical, we may have to encounter aberration of a lower order of small quantities, and therefore presumably of higher importance. By taking the axis of x and y coincident with the directions of principal curvature at O, we may write the equation of the surface z=^+ 1 '∕, + az 3 + βx 2 y + γxf + Sy 3 .... (2), p, p' being the principal radii of curvature, or focal lengths. The most important example of unsymmetrical aberration is in the spectroscope, where (if the faces of the prisms may be regarded as at any rate surfaces of revolution) the wave-surface may by suitable adjustments be rendered symmetrical with respect to the horizontal plane y= 0. This plane may then be regarded as primary, ρ being the primary focal length, at which distance the spectrum is formed. Under these circumstances β aud δ may be omitted from (2), which thus takes the form zy>2 Λ /2

z ^∕, + 2f + a ^ + yxy2 ( 3 >∙

The constants a and γ in (3) may be interpreted in terms of the differential coefficients of the principal radii of curvature. By the usual formula the radius of curvature at the point x of the intersection of (3) with the plane y=0 is approximately p(l - 6aρx). Since y = 0 is a principal plane throughout, this radius of curvature is a principal radius of the surface; so that, denoting it by p, we have

1 dp-ι a ~ø dx ^∙

Again, in the neighbourhood of the origin, the approximate value of the product of the principal curvatures is

1 ii>ax 2-yae

—7 d Γ^ Ί ·

PP P P

Thus

∕ 1 ∖ dp dp ’ 6ax 2yx , , x ‘ ∖pp'J fp p-p p' P ’ whence by (4)

1 dp'- 1 'y=⅛dx < 5 )∙

The equation of the normal at the point x, y, z is

-C- 2 = i-Ξ≡ = __!LZ# . (6)∙

- 1 p ^ 1 x + 3ctr 2 + yif f~ i y+ 2γit ’ y and its intersection with the plane ζ=ρ occurs at the point determined approximately by

ξ = - p(3cα 2 + γf∕ 2 ) )

,.⅛⅛⅛- 2w <'>> P ’ terms of the third order being omitted.

According to geometrical optics, the thickness of the image of a luminous line at the primary focus is determined by the extreme value of ξ; and for good definition in the spectroscope it is necessary to reduce this thickness as much as possible. One way of attaining the desired result would be to narrow the aperture; but, as we shall see later, to narrow the horizontal aperture is really to throw away the peculiar advantage of large instruments. The same objection, however, does not apply to narrowing the vertical aperture; and in many spectroscopes a great improvement in definition may be thus secured. In general, it is necessary that both γ and α be small. Since the value of ξ does not depend on p', it would seem that in respect of definition there is no advantage in avoiding astigmatism.

The width of the image when η = 0 (corresponding to y = 0) is 3α ρx 2 , and vanishes when α = 0, i.e., when there is no aberration for rays in the primary plane. In this case the image reduces to a linear arc. If further γ = 0, this arc becomes straight, and then the image at the primary focus is perfect to this order of approximation. As an example where α = 0, the image of a luminous point, formed at an equal distance on the further side of a sloped equi-convex lens, may be mentioned.

At the secondary focus, ζ=ρ', and from (6)

η=-2ρ'γxy (8).

If γ=0, the secondary focal line is formed without aberration, but not otherwise. Both focal lines are well formed when parallel rays fall upon a plano-convex lens, sloped at about 30°, the curved side of the lens being turned towards the parallel rays.

§ 7. Interference Fringes.

We have seen (§ 2) that, when two trains of parallel waves of equal wave-length are superposed, the intensity of the resultant depends upon the phase-relation of the components; but it is necessarily the same at all points of the wave-front. It not unfrequently happens that the parallelism of the component trains is approximate only, and there then arises the phenomenon known as interference fringes. If the two directions of propagation be inclined on opposite sides to the axis of x at small angles a, the expressions for two components of equal amplitudes are cos — < Vi - x cos a - y sin a ( , λ ( ) and cos ~ J Vi-Xcosa+ y sin a; so that the resultant is expressed by

„ 2πf∕ silla 2ιτ , λt . , , 1 .

2 cos— cos — (Vi-a: cos a) . . . . (1);

λ λ

from which it appears that the vibrations advance parallel to the axis of x, unchanged in type, and with a uniform velocity ^V∕eosa. Considered as depending on y, the vibration is a maximum when τ∕sinα is equal to 0, λ, 2λ, 3λ, &c., corresponding to the centres of the bright bands, while for the intermediate values ½λ, 3 / 2 λ, &c., there is no vibration. This is the interference of light proceeding from two similar homogeneous and very distant sources.

In the form of experiment adopted by Fresnel the sources O 1 , O 2 are situated at a finite distance D from the place of observation (Light, vol. xiv. p. 606). If A be the point of the screen equidistant from O 1 , O 2 , and P a neighbouring point, then approximately

O 1 P - O 3 P = √{D 2 + (u + ⅜δ) 2 } - √{ D 2 ÷ (« - ⅛δ) 2 } = ubfD, where O 1 O 2 =δ, AP=w.

Thus, if λ be the wave-length, the places where the phases are accordant are given by u≈nλD∕b (2), n being an integer.

If the light were really homogeneous, the successive fringes would be similar to one another and unlimited in number; moreover there would be no place that could be picked out by inspection as the centre of the system. In practice λ varies, and the only place of complete accordance for all kinds of light is at A, where u=0. Theoretically, there is no place of complete discordance for all kinds of light, and consequently no complete blackness. In consequence, however, of the fact that the range of sensitiveness of the eye is limited to less than an “octave,” the centre of the first dark band (on either side) is sensibly black, even when white light is employed; but it should be carefully remarked that tho existence of even one band is due to selection, and that the formation of several visible bands is favoured by the capability of the retina to make chromatic distinctions within the visible range.

The number of perceptible bands increases pari passu with the approach of the light to homogeneity. For this purpose there are two methods that may be used.

We may employ light, such as that from the soda flame, which possesses ab initio a high degree of homogeneity. If the range of wave-length included be a corresponding number of inter ference fringes may be made visible. The above is the number obtained by Fizeau, and Michelson has recently gone as far as 200,000. The narrowness of the bright line of light seen in the spectroscope, and the possibility of a large number of Fresnel’s bands, depend upon precisely the same conditions; the one is in truth as much an interference phenomenon as the other.

In the second method the original light may be highly composite, and homogeneity is brought about with the aid of a spectroscope. The analogy with the first method is closest if we use the spectroscope to give us a line of homogeneous light in simple substitution for the artificial flame. Or, following Foucault and Fizeau, we may allow the white light to pass, and subsequently analyse the mixture transmitted by a narrow slit in the screen upon which the interference bands are thrown. In the latter case we observe a channelled spectrum, with maxima of brightness corresponding to the wave-lengths bu ∕(n D). In either case the number of bands observable is limited solely by the resolving power of the spectroscope (§ 13), and proves nothing with respect to the regularity, or otherwise, of the vibrations of the original light.

The truth of this remark is strikingly illustrated by the possible formation, with white light, of a large number of achromatic bands. The unequal widths of the bands for the various colours, and consequent overlapping and obliteration, met with in the usual form of the experiment, depend upon the constancy of b (the mutual distance of the two sources) while λ varies. It is obvious that, if b were proportional to λ, the widths of the bands would be independent of λ, and that the various systems would fit together perfectly. To carry out the idea in its entirety, it would be necessary to use a diffraction spectrum as a source, and to duplicate this by Lloyd’s method with a single reflector placed so that b = 0 when λ = 0. In practice a sufficiently good result could doubtless be obtained with a prismatic spectrum (especially if the red and violet were removed by absorbing agents) under the condition that d (b ∕λ) = 0 in the yellow-green. It is remarkable that, in spite of the achromatic character of the bauds, their possible number is limited still by the resolving power of the instrument used to form the spectrum.

If a system of Fresnel’s bands be examined through a prism, the central white band undergoes an abnormal displacement, which has been supposed to be inconsistent with theory. The explanation has been shown by Airy to depend upon the peculiar manner in which the white band is in general formed.

“Any one of the kinds of homogeneous light composing the incident heterogeneous light will produce a series of bright and dark bars, unlimited in number as far as the mixture of light from the two pencils extends, and undistinguishable in quality. The consideration, therefore, of homogeneous light will never enable us to determine which is the point that the eye immediately turns to as the centre of the fringes. What then is the physical circumstance that determines the centre of the fringes?

“The answer is very easy. For different colours the bars have different breadths. If then the bars of all colours coincide at one part of the mixture of light, they will not coincide at any other part; but at equal distances on both sides from that place of coincidence they will be equally far from a state of coincidence. If then we can find where the bars of all colours coincide, that point is the centre of the fringes.

“It appears then that the centre of the fringes is not necessarily the point where the two pencils of light have described equal paths, but is determined by considerations of a perfectly different kind The distinction is important in this and in other experiments.”

The effect in question depends upon the dispersive power of the prism. If v be the linear shifting due to the prism of the originally central band, v must be regarded as a function of λ. Measured from the original centre, the position of the n th bar is now ν+nλD∣b.

The coincidence of the various bright bands occurs when this quantity is as independent as possible of λ, that is, when n is the nearest integer to ,t ~~D(7λ (3); or, as Airy expresses it in terms of the width of a band (h ), n = - dv∣dh.

The apparent displacement of the white band is thus not v simply, but dv v ~ k dh w '

The signs of dυ and dh being opposite, the abnormal displacement is in addition to the normal effect of the prism. But, since dυ/dh, or dv∣dλ, is not constant, the achromatism of the white band is less perfect than when no prism is used.

If a grating were substituted for the prism, v would vary as h, and (4) would vanish, so that in all orders of spectra the white band would be seen undisplaced.

The theoretical error, dependent upon the dispersive power, involved in the method of determining the refractive index of a plate by means of the displacement of a system of interference fringes (Light, vol. xiv. p. 607) has been discussed by Stokes. In the absence of dispersion the retardation R due to the plate would be independent of λ, and therefore completely compensated at the point determined by u = DR∕ b ; but when there is dispersion it is accompanied by a fictitious displacement of the fringes on the principle explained by Airy.

More recently the matter has engaged the attention of Cornu, who thus formulates the general principle :—" Dans un système de franges d'interférences produites à l'aide d’une lumière hétérogène ayant un spectre continu, il existe toujours une frange achromatique qui joue le rôle de frange centrale et qui se trouve au point de champ où les radiations les plus intenses présentent une différence de phase maximum ou minimum.”

In Fresnel’s experiment, if the retardation of phase due to an interposed plate, or to any other cause, be F(λ), the whole relative retardation of the two pencils at the point u is

≠=F(λ) + ⅛ (5); and the situation of the central, or achromatic, band is determined, not by φ = 0, but by dφ∣dκ == 0, or u = λ 2 DF(λ)∕δ (6).

In the theoretical statement we have supposed the source of light to be limited to a mathematical point, or to be extended only in the vertical direction (parallel to the bauds). Such a vertical extension, while it increases illumination, has no prejudicial effect upon distinctness, the various systems due to different points of the luminous line being sensibly superposed. On the other hand, the horizontal dimension of the source must be confined within narrow limits, the condition obviously being that the displacement of the centre of the system incurred by using in succession the two edges only of the slit should be small in comparison with the width of an interference band.

Before quitting this subject it is proper to remark that Fresnel’s bands are more influenced by diffraction than their discoverer supposed. On this account the fringes are often unequally broad and undergo fluctuations of brightness. A more precise calculation has been given by H. F. Weber and by H. Struve, but the matter is too complicated to be further considered here. The observations of Struve appear to agree well with the corrected theory.

§ 8. Colours of Thin Plates.

When plane waves of homogeneous light (λ) fall upon a parallel plate of index μ, the resultant reflected wave is made up of an infinite number of components, of which the most important are the first, reflected at the upper surface of the plate, and the second, transmitted at the upper surface, reflected at the under surface, and then transmitted at the upper surface. It is readily proved (Light, vol. xiv. p. 608) that so far as it depends upon the distances to be travelled in the plate and in air the retardation (δ) of the second wave relatively to the first is given by

δ = 2,u<cosα' (1), where t denotes the thickness of the plate, and α' the angle of refraction corresponding to the first entrance. If we represent all the vibrations by complex quantities, from which finally the imaginary parts are to be rejected, the retardation δ may be expressed by the introduction of the factor e ~ iκ8 , where i= ∖∕( - 1), and κ = c lπ∣κ.

At each reflexion or refraction the amplitude of the incident wave must be supposed to be altered by a certain factor. When the light proceeds from the surrounding medium to the plate, the factor for reflexion will be supposed to be b, and for refraction c; the corresponding quantities when the progress is from the plate to the surrounding medium will be denoted by e, f. Denoting the incident vibration by unity, we have then for the first component of the reflected wave b, for the second cefe~ iκ ^, for the third affρ~ 2iκ5 , and so on. Adding these together, and summing the geometric series, we find

(2)∙ l- c 2 e -uδ

In like manner for the λvave transmitted through the plate we get cf l-e* f ~ iκs

The quantities b, c, e, f are not independent. The simplest way to find the relations between them is to trace the consequences of supposing δ = 0 in (2) and (3). For it is evident a priori that with a plate of vanishing thickness there must be a vanishing reflexion, and a total transmission. Accordingly, b + e = 0, cf =1- e 2 (4), the first of which embodies Arago’s law of the equality of reflexions, as well as the famous “loss of half an undulation.” Using these we find for the reflected vibration, l-Λ- fat lλ and for the transmitted vibration l-c 2

1 - e 2 ∈ -iκδ

The intensities of the reflected and transmitted lights are the squares of the moduli of these expressions. Thus

Intensity of reflected light = e 2 -· (1 ; cos κ fζi + sιn j o (1-c 2 cos κδ) 2 + e 4 sin 2 κδ

= 4e 2 sin 2 (⅛⅝δ)

1 - 2e 2 cos κS + e i ’ · ’

Intensity of transmitted light = -c f- . . . , g . σ 1 — 2e 2 cos κδ + e 4 '

the sum of the two expressions being unity.

According to (7) not only does the reflected light vanish completely when δ = 0, but also whenever ½kδ = nπ, n being an integer, that is, whenever δ = n λ. When the first and third medium are the same, as we have here supposed, the central spot in the system of Newton’s rings is black, even though the original light contain a mixture of all wave lengths. The general explanation of the colours of Newton’s rings is given under Light, to which reference must be made. If the light reflected from a plate of any thickness be examined with a spectroscope of sufficient resolving power (§ 13), the spectrum will be traversed by dark bands, of which the centre corresponds to those wave lengths which the plate is incompetent to reflect. It is obvious that there is no limit to the fineness of the bands which may be thus impressed upon a spectrum, whatever may be the character of the original mixed light.

The relations between the factors b, c, e, f have been proved, independently of the theory of thin plates, in a general manner by Stokes, who called to his aid the general mechanical principle of reversibility. If the motions constituting the reflected and refracted rays to which an incident ray gives rise be supposed to be reversed, they will reconstitute a reversed incident ray. This gives one relation; and another is obtained from the consideration that there is no ray in the second medium, such as would be generated by the operation alone of either the reversed reflected or refracted rays. Space does not allow of the reproduction of the argument at length, but a few words may perhaps give the reader an idea of how the conclusions are arrived at. The incident ray (IA) being 1, the reflected (AR) and refracted (AF) rays are denoted by b and c. When b is reversed, it gives rise to a reflected ray b 2 along AI, and a refracted ray bc along AG (say).

When c is reversed, it gives rise to ef along AI, and ce along AG. Hence bc + ce = 0, b 2 + cf = 1, which agree with (4).

It is here assumed that there is no change of phase in the act of reflexion or refraction, except such as can be represented by a change of sign. Professor Stokes has, however, pushed the application of his method to the case where changes of phase are admitted, and arrives at the conclusion that “the sum of the accelerations of phase at the two reflexions is equal to the sum of the accelerations at the two refractions, and the accelerations of the two refractions are equal to each other.” The accelerations are supposed to be so measured as to give like signs to c and f, and unlike to b and e. The same relations as before obtain between the factors b, c, e, f expressing the ratios of amplitudes.

When the third medium differs from the first, the theory of thin plates is more complicated, and need not here be discussed. One particular case, however, may be mentioned. When a thin transparent film is backed by a perfect reflector, no colours should be visible, all the light being ultimately reflected, whatever the wave length may be. The experiment may be tried with a thin layer of gelatin on a polished silver plate. In other cases where a different result is observed, the inference is that either the metal does not reflect perfectly, or else that the material of which the film is composed is not sufficiently transparent.

Theory and observation alike show that the transmitted colours of a thin plate, e.g., a soap film or a layer of air, arc very inferior to those reflected. Specimens of ancient glass, which have undergone superficial decomposition, on the other hand, sometimes show transmitted colours of remarkable brilliancy. The probable explanation, suggested by Brewster, is that we have here to deal not merely with one, but with a series of thin plates of not very different thicknesses. It is evident that with such a series the transmitted colours would be much purer, and the reflected much brighter, than usual. If the thicknesses are strictly equal, certain wave-lengths must still be absolutely missing in the reflected light; while on the other hand a constancy of the interval between the plates will in general lead to a special preponderance of light of some other wave-length for which all the component parts as they ultimately emerge are in agreement as to phase.

All that can be expected from a physical theory is the determination of the composition of the light reflected from or transmitted by a thin plate in terms of the composition of the incident light. The further question of the chromatic character of the mixtures thus obtained belongs rather to physiological optics, and cannot be answered without a complete knowledge of the chromatic relations of the spectral colours themselves. Experiments upon this subject have been made by various observers, and especially by Maxwell, who has exhibited his results on a colour diagram as used by Newton. A calculation of the colours of thin plates, based upon Maxwell’s data, and accompanied by a drawing showing the curve representative of the entire series up to the fifth order, has recently been published; and to this the reader who desires further information must be referred, with the remark that the true colours are not seen in the usual manner of operating with a plate of air enclosed between glass surfaces, on account of the contamination with white light reflected at the other surfaces of the glasses. This objection is avoided when a soap film is employed, to the manifest advantage of the darker colours, such as the red of the first order. The colours of Newton’s scale are met with also in the light transmitted by a somewhat thin plate of doubly-refracting material, such as mica, the plane of analysis being perpendicular to that of primitive polarization.

The same series of colours occur also in other optical experiments, e.g., at the centre of the illuminated area when light issuing from a point passes through a small round aperture in an otherwise opaque screen (§ 10).

The colours of which we have been speaking are those formed at nearly perpendicular incidence, so that the retardation (reckoned as a distance), viz., 2μ t cos α', is sensibly independent of λ. This state of things may be greatly departed from when the thin plate is rarer than its surroundings, and the incidence is such that a' is nearly equal to 90°, for then, in consequence of the powerful dispersion, cos a ' may vary greatly as we pass from one colour to another. Under these circumstances the series of colours entirely alters its character, and the bands (corresponding to a graduated thickness) may even lose their coloration, becoming sensibly black and white through many alternations. The general explanation of this remarkable phenomenon was suggested by Newton, but it does not appear to have been followed out in accordance with the wave theory.

Let us suppose that plane waves of white light travelling in glass are incident at angle a upon a plate of air, which is bounded again on the other side by glass. If μ be the index of the glass, a the angle of refraction, then sinα' = μsinα; and the retardation, expressed by the equivalent distance in air, is 2t sec a' - μ . 2t tan a' sin a = 2t cos a'; and the retardation in phase is 2tcosα'∕λ , λ being as usual the wavelength in air.

The first thing to be noticed is that, when α approaches the critical angle, cos α' becomes as small as we please, and that consequently the retardation corresponding to a given thickness is very much less than at perpendicular incidence. Hence the glass surfaces need not be so close as usual.

A second feature is the increased brilliancy of the light. According to (7) the intensity of the reflected light when at a maximum (sin½κδ = 1) is 4 e 2 ∕(1 + e 2 ) 2 . At perpendicular incidence e is about ⅕, and the intensity is somewhat small; but, as cos a' approaches zero, e approaches unity (§ 2G), and the brilliancy is much increased.

But the peculiarity which most demands attention is the lessened influence of a variation in λ upon the phase-retardation. A diminution of λ of itself increases the retardation of phase, but, since waves of shorter wave-length are more refrangible, this effect may be more or less perfectly compensated by the greater obliquity, and consequent diminution in the value of cos a'. We will investigate the conditions under which the retardation of phase is stationary in spite of a variation of λ.

In order that λ -1 cos α ' may be stationary, we must have λsin α'dα ' + cos a' d λ = 0 , where (a being constant)

cos a'da' = sin a dμ.

Thus cot 2 a'==- — ⅛- (9), μdK y , giving a' when the relation between μ and λ is known.

According to Cauchy’s formula, which represents the facts very well throughout most of the visible spectrum,

μ-A + Bλ- 2 (10), so that cot 2 α'=⅞- = 2 ^~-) (ll),

λ¼ μ

If we take, as for Chance’s “extra-dense flint,” B= ’ 984 × 10* 10 , and, as for the soda lines, μ = l - 65, λ = 5 ’ 89×10^ 5 , we get

α' = 79° 30'.

At this angle of refraction, and with this kind of glass, the retardation of phase is accordingly nearly independent of wave-length, and therefore the bands formed, as the thickness varies, are approximately achromatic. Perfect achromatism would be possible only under a law of dispersion

μ 2 =A'-B'λ s .

If the source of light be distant aud very small, the black bands are wonderfully fine and numerous. The experiment is best made (after Newton) with a right-angled prism, whose hypothenusal surface may be brought into approximate contact with a plate of black glass. The bands should be observed with a convex lens, of about 8 inches focus. If the eye be at twice this distance from the prism, and the lens be held midway between, the advantages are combined of a large field and of maximum distinctness.

If Newton’s rings are examined through a prism, some very remarkable phenomena are exhibited, described in his twenty-fourth observation. “When the two object-glasses are laid upon one another, so as to make the rings of the colours appear, though with my naked eye I could not discern above eight or nine of those rings, yet by viewing them through a prism I could see a far greater multitude, insomuch that I could number more than forty And I

believe that the experiment may be improved to the discovery of far greater numbers But it was on but one side of these rings, namely, that towards which the refraction was made, which by the refraction was rendered distinct, and the other side became more confused than when viewed with the naked eye

“I have sometimes so laid one object-glass upon the other that to the naked eye they have all over seemed uniformly white, without the least appearance of any of the coloured rings; and yet by viewing them through a prism great multitudes of those rings have discovered themselves.”

Newton was evidently much struck with these “so odd circumstances;” and he explains the occurrence of the rings at unusual thicknesses as due to the dispersing power of the prism. The blue system being more refracted than the red, it is possible under certain conditions that the n th blue ring may be so much displaced relatively to the corresponding red ring as at one part of the circumference to compensate for the different diameters. A white stripe may thus be formed in a situation where without the prism the mixture of colours would be complete, so far as could be judged by the eye.

The simplest case that can be considered is when the “thin plate” is bounded by plane surfaces inclined to one another at a small angle. By drawing back the prism (whose edge is parallel to the intersection of the above-mentioned planes) it will always be possible so to adjust the effective dispersing power as to bring the n th bars to coincidence for any two assigned colours, and therefore approximately for the entire spectrum. The formation of the achromatic band, or rather central black band, depends indeed upon the same principles as the fictitious shifting of the centre of a system of Fresnel’s bands when viewed through a prism.

But neither Newton nor, as would appear, any of his successors has explained why the bands should be more numerous than usual, and under certain conditions sensibly achromatic for a large number of alternations. It is evident that, in the particular case of the wedge-shaped plate above specified, such a result would not occur. The width of the bands for any colour would be proportional to λ, as well after the displacement by the prism as before; and the succession of colours formed in white light and the number of perceptible bands would be much as usual.

The peculiarity to be explained appears to depend upon the curvature of the surfaces bounding the plate. For simplicity suppose that the lower surface is plane (y =0), and that the approximate equation of the upper surface is y=α + δsc 2 , a being thus the least distance between the plates. The black of the n th order for wavelength λ occurs when

⅜nλ-α + ⅛z 3 (12); ami thus the width (Sa?) at this place of the band is given by

⅛λ = 25zδz (13), or 5 ' c = ‰ = 4√V. √(⅜7iλ-αj (U )·

If the glasses be in contact, as is usually supposed in the theory of Newton’s rings, a=0. and δA ’ ccλ i , or the width of the band of the n th order varies as the square root of the wave-length, instead of as the first power. Even in this case the overlapping and subsequent obliteration of the bands is greatly retarded by the use of the prism, but the full development of the phenomenon requires that a should be finite. Let us inquire what is the condition in order that the width of the band of the n th order may be stationary, as λ varies. By (14) it is necessary that the variation of λ 2 ∕(½ n λ- a ) should vanish. Hence a=¼nλ, so that the interval between the surfaces at the place where the n th band is formed should be half due to curvature and half to imperfect contact at the place of closest approach. If this condition be satisfied, the achromatism of the n th band, effected by the prism, carries with it the achromatism of a large number of neighbouring bands, and thus gives rise to the remarkable effects described by Newton.

§ 9. Newton’s Diffusion Rings.

In the fourth part of the second book of his Optics Newton investigates another series of rings, usually (though not very appropriately) known as the colours of thick plates. The fundamental experiment is as follows. At the centre of curvature of a concave looking-glass, quicksilvered behind, is placed an opaque card, perforated by a small hole through which sunlight is admitted. The main body of the light returns through the aperture; but a series of concentric rings are seen upon the card, the formation of which was proved by Newton to require the co-operation of the two surfaces of the mirror. Thus the diameters of the rings depend upon the thickness of the glass, and none are formed when the glass is replaced by a metallic speculum. The brilliancy of the rings depends upon imperfect polish of the anterior surface of the glass, and may be augmented by a coat of diluted milk, a device used by the Duc de Chaulnes. The rings may also be well observed without a screen in the manner recommended by Stokes. For this purpose all that is required is to place a small flame at the centre of curvature of the prepared glass, so as to concide with its image. The rings are then seen surrounding the flame and occupying a definite position in space.

The explanation of the rings, suggested by Young, and developed by Herschel, refers them to interference between one portion of light scattered or diffracted by a particle of dust, and then regularly refracted and reflected, and another portion first regularly refracted and reflected and then diffracted at emergence by the same particle. It has been shown by Stokes that no regular interference is to be expected between portions of light diffracted by different particles of dust.

In the memoir of Stokes will be found a very complete discussion of the whole subject, and to this the reader must be referred who desires a fuller knowledge. Our limits will not allow us to do more than touch upon one or two points. The condition of fixity of the rings when observed in air, and of distinctness when a screen is used, is that the systems due to all parts of the diffusing surface should concide; and it is fulfilled only when, as in Newton’s experiments, the source and screen are in the plane passing through the centre of curvature of the glass.

As the simplest for actual calculation, we will consider a little further the case where the glass is plane and parallel, of thickness t and index μ, and is supplemented by a lens at whose focus the source of light is placed. This lens acts both as collimator aud as object-glass, so that the combination of lens and plane mirror replaces the concave mirror of Newton’s experiment. The retardation is calculated in the same way as for thin plates. In fig. 2 the diffracting particle is situated at B, and we have to find the relative retardation of the two rays which emerge finally at inclination θ, the one diffracted at emergence following the path ABDBIE, and the other diffracted at entrance and following the path ABFGH. The retardation of the former from B to I is 2μ + BI, and of the latter from B to the equivalent place G is 2μBF. Now FB=√sec0', 0' being the angle of refraction; BI = 2Ztan0'sin0; so that the relative retardation R is given by

R = 2μ(!{l + μ^ 1 tan θ' sin 0 -sec 0'} = 2μi(l - cos 0 ’).

If 0,0' be small, we may take

R = 2Z0 2 ∕ μ (1), as sufficiently approximate.

The condition of distinctness is here satisfied, since R is the same for every ray emergent parallel to a given one. The rays of one parallel system are collected by the lens to a focus at a definite point in the neighbourhood of the original source.

The formula (1) was discussed by Herschel, and shown to agree with Newton’s measures. The law of formation of the rings follows immediately from the expression for the retardation, the radius of the ring of n th order being proportional to n and to the square root of the wave-length.

§10. Huygens's Principle. Theory of Shadows.

The objection most frequently brought against the undulatory theory in its infancy was the difficulty of explaining in accordance with it the existence of shadows. Thanks to Fresnel and his followers, this department of optics is now precisely the one in which the theory has secured its greatest triumphs.

The principle employed in these investigations is due to Huygens, and may be thus formulated. If round the origin of waves an ideal closed surface be drawn, the whole action of the waves in the region beyond may be regarded as due to the motion continually propagated across the various elements of this surface. The wave motion due to any element of the surface is called a secondary wave, and in estimating the total effect regard must be paid to the phases as well as the amplitudes of the components. It is usually convenient to choose as the surface of resolution a waυe-front, i.e., a surface at which the primary vibrations are in one phase.

Any obscurity that may hang over Huygens’s principle is due mainly to the indefiniteness of thought and expression which we must be content to put up with if we wish to avoid pledging ourselves as to the character of the vibrations. In the application to sound, where we know what we are dealing with, the matter is simple enough in principle, although mathematical difficulties would often stand in the way of the calculations we might wish to make. The ideal surface of resolution may be there regarded as a flexible lamina; and we know that, if by forces locally applied every element of the lamina be made to move normally to itself exactly as the air at that place does, the external aerial motion is fully determined. By the principle of superposition the whole effect may be found by integration of the partial effects due to each element of the surface, the other elements remaining at rest.

We will now consider in detail the important case in which uniform plane waves are resolved at a surface coincident with a wave-front (OQ). We imagine the wave-front divided into elementary rings or zones, called Huygens’s zones, by spheres described round P (the point at which the aggregate effect is to be estimated), the first sphere, touching the plane at O, with a radius equal to PO, and the succeeding spheres with radii increasing at each step by ½λ. There are thus marked out a series of circles, whose radii x are given by x 2 + r 2 = (r + ½ n λ) 2 , or x 2 =nλr nearly; so that the rings are at first of nearly equal area. Now the effect upon P of each element of the plane is proportional to its area; but it depends also upon the distance from P, and possibly upon the inclination of the secondary ray to the direction of vibration and to the wave-front. These questions will be further considered in connexion with the dynamical theory; but under all ordinary circumstances the result is independent of the precise answer that may be given. All that it is necessary to assume is that the effects of the successive zones gradually diminish, whether from the increasing obliquity of the secondary ray or because (on account of the limitation of the region of integration) the zones become at last more and more incomplete. The component vibrations at P due to the successive zones are thus nearly equal in amplitude and opposite in phase (the phase of each corresponding to that of the infinitesimal circle midway between the boundaries), and the series which we have to sum is one in which the terms are alternately opposite in sign and, while at first nearly constant in numerical magnitude, gradually diminish to zero. In such a series each term may be regarded as very nearly indeed destroyed by the halves of its immediate neighbours, and thus the sum of the whole series is represented by half the first term, which stands over uncompensated. The question is thus reduced to that of finding the effect of the first zone, or central circle, of which the area is πλ r.

We have seen that the problem before us is independent of the law of the secondary wave as regards obliquity; but the result of the integration necessarily involves the law of the intensity and phase of a secondary wave as a function of r, the distance from the origin. And we may in fact, as was done by A. Smith, determine the law of the secondary wave, by comparing the result of the integration with that obtained by supposing the primary wave to pass on to P without resolution.

Now as to the phase of the secondary wave, it might appear natural to suppose that it starts from any point Q with the phase of the primary wave, so that on arrival at P it is retarded by the amount corresponding to QP. But a little consideration will prove that in that case the series of secondary waves could not reconstitute the primary wave. For the aggregate effect of the secondary waves is the half of that of the first Huygens zone, and it is the central element only of that zone for which the distance to be travelled is equal to r. Let us conceive the zone in question to be divided into infinitesimal rings of equal area. The effects due to each of these rings are equal in amplitude and of phase ranging uniformly over half a complete period. The phase of the resultant is midway between those of the extreme elements, that is to say, a quarter of a period behind that due to the element at the centre of the circle. It is accordingly necessary to suppose that the secondary waves start with a phase one-quarter of a period in advance of that of the primary wave at the surface of resolution.

Further, it is evident that account must be taken of the variation of phase in estimating the magnitude of the effect at P of the first zone. The middle element alone contributes without deduction; the effect of every other must be found by introduction of a resolving factor, equal to cos θ , if θ represent the difference of phase between this element and the resultant. Accordingly, the amplitude of the resultant will be less than if all its components had the same phase, in the ratio

∕-+i^

∕ C0sθllθ'.1Γ , J-∖π or 2:π. Now 2 area ∕π = 2λ r ; so that, in order to reconcile the amplitude of the primary wave (taken as unity) with the half effect of the first zone, the amplitude, at distance r, of the secondary wave emitted from the element of area dS must be taken to be

''⅛ (i).

λr

By this expression, in conjunction with the quarter-period acceleration of phase, the law of the secondary wave is determined.

That the amplitude of the secondary wave should vary as r -1 was to be expected from considerations respecting energy; but the occurrence of the factor λ -1 , and the acceleration of phase, have sometimes been regarded as mysterious. It may be well therefore to remember that precisely these laws apply to a secondary wave of sound, which can be investigated upon the strictest mechanical principles.

The recomposition of the secondary waves may also be treated analytically. If the primary wave at O be cos κat, the effect of the secondary wave proceeding from the element dS at Q is dS , 1 1 . dS . . t ,

—cosκ(α<-p + ⅛λ)= - — sin κ(αt - p). λp λp

If √S = 27τ∙r⅛ we have for the whole effect

2π ∕^ 3θ sin κ{at - p) xdx

λ√o P or, since xdx=pdp. κ = 2∙∣r∣λ, - κJ~ r c °βin κ(at - p)dp = [ - cos κ(at - p)] r .

In order to obtain the effect of the primary wave, as retarded by traversing the distance r, viz., cos κ (at — r), it is necessary to suppose that the integrated term vanishes at the upper limit. And it is important to notice that without some further understanding the integral is really ambiguous. According to the assumed law of the secondary wave, the result must actually depend upon the precise radius of the outer boundary of the region of integration, supposed to be exactly circular. This case is, however, at most very special and exceptional. We may usually suppose that a large number of the outer rings are incomplete, so that the integrated term at the upper limit may properly be taken to vanish. If a formal proof be desired, it may be obtained by introducing into the integral a factor such as c~ i, p, in whieh h is ultimately made to diminish without limit.

When the primary wave is plane, the area of the first Huygens zoneisπλr, and, since the secondary waves vary as r^ 1 , theintensity is independent of r, as of course it should be. If, however, the primary wave be spherical, and of radius a at the wave-front of resolution, then we know that at a distance r further on the amplitude of the primary wave will be diminished in the ratio α∙.(r + α). This may be regarded as a consequence of the altered area of the first Huygens zone. For, if x be its radius, we have

{(r+⅛λ) 2 - x 2 } + √{α 2 -ie 2 }=r + α, so that

„ λαr . a“ = nearly.

a + r j

Since the distance to be travelled by the secondary waves is still r, we see how the effect of the first zone, and therefore of the whole series is proportional toα∕(α + r). In like manner may be treated other cases, such as that of a primary wave-front of unequal principal curvatures.

The general explanation of the formation of shadows may also be conveniently based upon Huygens’s zones. If the point under consideration be so far away from the geometrical shadow that a large number of the earlier zones are complete, then the illumination, determined sensibly by the first zone, is the same as if there were no obstruction at all. If, on the other hand, the point be well immersed in the geometrical shadow, the earlier zones are altogether missing, and, instead of a series of terms beginning with finite numerical magnitude and gradually diminishing to zero, we have now to deal with one of which the terms diminish to zero at both ends. The sum of such a series is very approximately zero, each term being neutralized by the halves of its immediate neighbours, which are of the opposite sign. The question of light or darkness then depends upon whether the series begins or ends abruptly. With few exceptions, abruptness can occur only in the presence of the first term, viz., when the secondary wave of least retardation is unobstructed, or when a ray passes through the point under consideration. According to the undulatory theory the light cannot be regarded strictly as travelling along a ray; but the existence of an unobstructed ray implies that the system of Huygens’s zones can be commenced, and, if a large number of these zones are fully developed and do not terminate abruptly, the illumination is unaffected by the neighbourhood of obstacles. Intermediate cases in which a few zones only are formed belong especially to the province of diffraction.

An interesting exception to the general rule that full brightness requires the existence of the first zone occurs when the obstacle assumes the form of a small circular disk parallel to the plane of the incident waves. In the earlier half of the 18th century Delisle found that the centre of the circular shadow was occupied by a bright point of light, but the observation passed into oblivion until Poisson brought forward as an objection to Fresnel’s theory that it required at the centre of a circular shadow a point as bright as if no obstacle were intervening. If we conceive the primary wave to be broken up at the plane of the disk, a system of Huygens’s zones can be constructed which begin from the circumference; and the first zone external to the disk plays the part ordinarily taken by the centre of the entire system. The whole effect is the half of that of the first existing zone, and this is sensibly the same as if there were no obstruction.

When light passes through a small circular or annular aperture, the illumination at any point along the axis depends upon the precise relation between the aperture and the distance from it at which the point is taken. If, as in the last paragraph, we imagine a system of zones to be drawn commencing from the inner circular boundary of the aperture, the question turns upon the manner in which the series terminates at the outer boundary. If the aperture be such as to fit exactly an integral number of zones, the aggregate effect may be regarded as the half of those due to the first and last zones. If the number of zones be even, the action of the first and last zones are antagonistic, and there is complete darkness at the point. If on the other hand the number of zones be odd, the effects conspire; and the illumination (proportional to the square of the amplitude) is four times as great as if there were no obstruction at all.

The process of augmenting the resultant illumination at a particular point by stopping some of the secondary rays may be carried much further. By the aid of photography it is easy to prepare a plate, transparent where the zones of odd order fall, and opaque where those of even order fall. Such a plate has the power of a condensing lens, and gives an illumination out of all proportion to what could be obtained without it. An even greater effect (fourfold) would be attained if it were possible to provide that the stoppage of the light from the alternate zones were replaced by a phase-reversal without loss of amplitude.

In such experiments the narrowness of the zones renders necessary a pretty close approximation to the geometrical conditions. Thus in the case of the circular disk, equidistant (r) from the source of light and from the screen upon whieh the shadow is observed, the width of the first exterior zone is given by dx≈^ 4(2x) ’ 2x being the diameter of the disk. If 2r = 1000 cm., 2x=l cm., λ = 6×10^ 5 cm., then √z = '0015 cm. Hence, in order that this zone may be perfectly formed, there should be no error in the circumference of the order of '001 cm. The experiment succeeds in a dark room of the length above mentioned, with a threepenny bit (supported by three threads) as obstacle, the origin of light being a small needle hole in a plate of tin, through which the sun’s rays shine horizontally after reflexion from an external mirror. In the absence of a heliostat it is more convenient to obtain a point of light with the aid of a lens of short focus.

The amplitude of the light at any point in the axis, when plane waves are incident perpendicularly upon an annular aperture, is, as above, cos κ(μt - r 1 ) - cos κ(at - r 2 ) = 2 sin κat. sin κ(r 1 - 7∙ 2 ),

To, r 1 being the distances of the outer and inner boundaries from the point in question. It is scarcely necessary to remark that in all such cases the calculation applies in the first instance to homogeneous light, and that, in accordance with Fourier’s theorem, each homogeneous component of a mixture may be treated separately. When the original light is white, the presence of some components and the absence of others will usually give rise to coloured effects, variable with the precise circumstances of the case.

Although what we have to say upon the subject is better postponed until we consider the dynamical theory, it is proper to point out at once that there is an element of assumption in the application of Huygens’s principle to the calculation of the effects produced by opaque screens of limited extent. Properly applied, the principle could not fail; but, as may readily be proved in the case of sonorous waves, it is not in strictness sufficient to assume the expression for a secondary wave suitable when the primary wave is undisturbed, with mere limitation of the integration to the transparent parts of the screen. But, except perhaps in the case of very fine gratings, it is probable that the error thus caused is insignificant; for the incorrect estimation of the secondary waves will be limited to distances of a few wave-lengths only from the boundary of opaque and transparent parts.

§ ll. Fraunhofer’s Diffraction Phenomena.

A very general problem in diffraction is the investigation of the distribution of light over a screen upon which impinge divergent or convergent spherical waves after passage through various diffracting apertures. When the waves are convergent and the recipient screen is placed so as to contain the centre of convergency—the image of the original radiant point, the calculation assumes a less complicated form. This class of phenomena was investigated by Fraunhofer (upon principles laid down by Fresnel), and are sometimes called after his name. We may conveniently commence with them on account of their simplicity and great importance in respect to the theory of optical instruments.

If f be the radius of the spherical wave at the place of resolution, where the vibration is represented by cos nat, then at any point M (fig. 4) in the recipient screen the vibration due to an element dS of the wave-front is (§ 9)

dS . , , .

sin κ(tw - p) , λρ

p being the distance between M and the element dS.

Taking coordinates in the plane of the screen with the centre of the wave as origin, let us represent M by ξ, ∙η, and P (where dS is situated) by x, y, z.

Then p 2 =(^-ξ) 2 +(2∕-η) 3 +2 2 . f 2 =a≈ a +y a +≡ 2 ; so that p2 = y 2 _ 2 λ ∙ξ - 2y∙η + ξ 2 + η 2 .

In the applications with which we are concerned, ξ, ?; are very small quantities; and we may take

At the same time dS may be identified with dxdy, and in the denominator p may be treated as constant and equal to f. Thus the expression for the vibration at M becomes at-f+^~ dxdy . . . (1); and for the intensity, represented by the square of the amplitude, P*∙j⅛ J^wn∙κ c M~dxdy^ +^ 2 JJ∞**½^ dxd y∖ ( 2 )∙ This expression for the intensity becomes rigorously applicable when f is indefinitely great, so that ordinary optical aberration disappears. The incident waves are thus plane, and are limited to a plane aperture coincident with a wave-front. The integrals are then properly functions of the direction in which the light is to be estimated.

In experiment under ordinary circumstances it makes no difference whether the collecting lens is in front of or behind the diffracting aperture. It is usually most convenient to employ a telescope focused upon the radiant point, and to place the diffracting apertures immediately in front of the object-glass. What is seen through the eye-piece in any case is the same as would be depicted upon a screen in the focal plane.

Before proceeding to special cases it maybe well to call attention to some general properties of the solution expressed bv (2).

If, when the aperture is given, the wave-length (proportional to κ^ 1 ) varies, the composition of the integrals is unaltered, provided ξ and η are taken universely proportional to λ. A diminution of λ thus leads to a simple proportional shrinkage of the diffraction pattern, attended by an augmentation of brilliancy in proportion to A’ 2 .

If the wave-length remains unchanged, similar effects are produced by an increase in the scale of the aperture. The linear dimension of the diffraction pattem is inversely as that of the aperture, and the brightness at corresponding points is as the sqicare of the area of aperture. '

If the aperture and wave-length increase in the same proportion, the size and shape of the diffraction pattern undergo no change.

We will now apply the integrals (2) to the case of a rectangular aperture of width a parallel to x and of width b parallel to y. The limits of integration for x may thus be taken to be - ⅛α and + ⅜α, and for y to be - ⅜fr, + ∖b. We readily find (with substitution for κ of 2π∕λ) . 7

' ’ .,τtaf . 9 πbη o , 2 sιn 2 -^-- sιn-- 7r = fff A ∕..√

∕ 2 λ 2 ’ > 2 il 2 ξ 2 ‘ τr⅜ 2 lj 2 ' ’ as representing the distribution of light in the image of a mathematical point when the aperture is rectangular, as is often the case in spectroscopes.

The second and third factors of (3) being each of the form sin 2 u∕u 2 , we have to examine the character of this function. It vanishes when u=mπ, m being any whole number other than zero. When w=0, it takes the value unity. The maxima occur when u = tanis, (4), and then sin 2 w∕u 2 =cos 2 w ....... (5).

To calculate the roots of (5) we may assume u=(nι+⅜)ιr-y=U-y , where y is a positive quantity which is small when u is large Substituting this, we find cot y=U-y, whence

„ AΛ4.Aj.^ a 4. ∖ y 3 2 y 5 17 ∕

j u∖ U U 2 √ 3 15 315 ·

This equation is to be solved by successive approximation. It will readily be found that u-u- i ,-v-u-.-∣v-≈-5∣u->-^u->-. . (β).

In the first quadrant there is no root after zero, since tan u>u, and in the second quadrant there is none because the signs of u and tan u are opposite. The first root after zero is thus in the third quadrant, corresponding to m=1. Even in this case the series converges sufficiently to give the value of the root with considerable accuracy, while for higher values of m it is all that could be desired. The actual values of u/π (calculated in another manner by Schwerd)are 1⋅4303, 2⋅4590, 3⋅4709, 4⋅4747, 5⋅4818, 6⋅4844, &c.

Since the maxima occur when u=(m + ½)π nearly, the successive values are not very different from

4 4 4

9τr 2 ’ 25τr 2 ’ 49π 2 ’

The application of these results to (3) shows that the field is brightest at the centre ξ = 0, η = 0, viz., at the geometrical image. It is traversed by dark lines whose equations are ξ=mfκ∣a, ∙η = mfκib.

Within the rectangle formed by pairs of consecutive dark lines, and not far from its centre, the brightness rises to a maximum; but these subsequent maxima are in all cases much inferior to the brightness at the centre of the entire pattern (ξ = 0, η = 0).

By the principle of energy the illumination over the entire focal plane must be equal to that over the diffracting area; and thus, in accordance with the suppositions by which (3) was obtained, its value when integrated from ξ = — ∞ to ξ=+∞, and from η = - ∞ to η=+∞ should be equal to ab. This integration, employed originally by Kelland to determine the absolute intensity of a secondary wave, may be at once effected by means of the known formula

∕∙-t-≈ · 2 ∕*+ a ° ·

sin 2 ii , ∕ sin u 1 - i ~du = ∕ du=ιτ.

□ο u- J -∞ U

It will be observed that, while the total intensity is proportional to ab, the intensity at the focal point is proportional to a 2 b 2 . If the aperture be increased, not only is the total brightness over the focal plane increased with it, but there is also a concentration of the diffraction pattern. The form of (3) shows immediately that, if a and b be altered, the coordinates of any characteristic point in the pattern vary as a -1 and b -1 .

The contraction of the diffraction pattern with increase of aperture is of fundamental importance with reference to the resolving power of optical instruments. According to common optics, where images are absolute, the diffraction pattern is supposed to be infinitely small, and two radiant points, however near together, form separated images. This is tantamount to an assumption that λ is infinitely small. The actual finiteness of λ imposes a limit upon the separating or resolving power of an optical instrument.

This indefiniteness of images is sometimes said to be due to diffraction by the edge of the aperture, and proposals have even been made for curing it by causing the transition between the interrupted and transmitted parts of the primary wave to be less abrupt. Such a view of the matter is altogether misleading. What requires explanation is not the imperfection of actual images so much as the possibility of their being as good as we find them.

At the focal point (ξ = 0,η = 0) all the secondary waves agree in phase, and the intensity is easily expressed, whatever be the form of the aperture. From the general formula (2), if A be the area of aperture,

I 0 2 = A 2 ∕λ 2 ∕ 2 (7).

The formation of a sharp image of the radiant point requires that the illumination become insignificant when ξ, η attain small values, and this insignificance can only arise as a consequence of discrepancies of phase among the secondary waves from various parts of the aperture. So long as there is no sensible discrepancy of phase there can be no sensible diminution of brightness as compared with that to be found at the focal point itself. We may go further, and lay it down that there can be no considerable loss of brightness until the difference of phase of the waves proceeding from the nearest and furthest parts of the aperture amounts to ⅛λ.

When the difference of phase amounts to λ, we may expect the resultant illumination to be very much reduced. In the particular case of a rectangular aperture the course of things can be readily followed, especially if we conceive f to be infinite. In the direction (suppose horizontal) for which η = 0, ξ∕f=sin0 , the phases of the secondary waves range over a complete period when sinθ = λ∕α, and, since all parts of the horizontal aperture are equally effective, there is in this direction a complete compensation and consequent absence of illumination. When sin0 = ⅛λ∕α, the phases range one and a half periods, and there is revival of illumination. We may compare the brightness with that in the direction 0 = 0. The phase of the resultant amplitude is the same as that due to the central secondary wave, and the discrepancies of phase among the components reduce the amplitude in the proportion a- 3,r

1 ∕' + 2' 3 π √⅛ c °s≠⅜≡L

2

2 4

or -: 1; so that the brightness in this direction is °f the maximum at 0 = 0. In like manner we may find the illumination in any other direction, and it is obvious that it vanishes when sin 0 is any multiple of λ∕α.

The reason of the augmentation of resolving power with aperture will now be evident. The larger the aperture the smaller are the angles through which it is necessary to deviate from the principal direction in order to bring in specified discrepancies of phase—the more concentrated is the image.

In many cases the subject of examination is a luminous line of uniform intensity, the various points of which are to be treated as independent sources of light. If the image of the line be ξ = 0, the intensity at any point ξ, η of the diffraction pattern may be represented by .

Λ”,, α¾ δiu2 V

J-* (8) ’

λ 2 ∕ a the same law as obtains for a luminous point when horizontal directions are alone considered. The definition of a fine vertical line, and consequently the resolving power for contiguous vertical lines, is thus independent of the vertical aperture of the instrument, a law of great importance in the theory of the spectroscope.

The distribution of illumination in the image of a luminous line is shown by the curve ABC (fig. 5), representing the value of the function sin 2 u∕u 2 from u = Q to w = 2π. The part corresponding to negative values of u is similar, OA being a line of symmetry.

Let us now consider the distribution of brightness in the image of a double line whose components are of equal strength, and at such an angular interval that the central line in the image of one coincides with the first zero of brightness in the image of the other. In fig. 5 the curve of brightness for ono component is ABC, and for tho other OΛ ’ C ’; and the curve representing half the combined brightnesses is ΕΈΕ. The brightness (corresponding to B) midway between the two central points AA' is ’ 8106 of the brightness at the central points themselves. We may consider this to be about the limit of closeness at which there could be any decided ap>pearance of resolution, though doubtless an observer accustomed to his instrument would recognize the duplicity with certainty. The obliquity, corresponding to u = π, is such that the phases of the secondary waves range over a complete period, i.e., such that the projection of the horizontal aperture upon this direction is one wave-length. We conclude that a double line cannot be fairly resolved unless its components subtend an angle exceeding that subtended by the wave-length of light at a distance equal to the horizontal aperture. This rule is convenient on account of its simplicity; and it is sufficiently accurate in view of the necessary uncertainty as to what exactly is meant by resolution.

On the experimental confirmation of the theory of the resolving power of rectangular apertures, see Optics, vol. xvii. p. 807.

If the angular interval between the components of a double line be half as great again as that supposed in the figure, the brightness midway between is ‘1802 as against 1 - 0450 at the central lines of each image. Such a falling off in the middle must be more than sufficient for resolution. If the angle subtended by the components of a double line be twice that subtended by the wave-length at a distance equal to the horizontal aperture, the central bands are just clear of one another, and there is a line of absolute blackness in the middle of the combined images.

Since the limitation of the width of the central band in the image of a luminous line depends upon discrepancies of phase among the secondary waves, and since the discrepancy is greatest for the waves which come from the edges of the aperture, the question arises how far the operation of the central parts of the aperture is advantageous. If we imagine the aperture reduced to two equal narrow slits bordering its edges, compensation will evidently be complete when the projection on an oblique direction is equal to ⅛λ, instead of λ as for the complete aperture. By this procedure the width of the central band in the diffraction pattern is halved, and so far an advantage is attained. But, as will be evident, the bright bands bordering the central band are now not inferior to it in brightness; in fact, a band similar to the central band is reproduced an indefinite number of times, so long as there is no sensible discrepancy of phase in the secondary waves proceeding from the various parts of the same slit. Under these circumstances the narrowing of the band is paid for at a ruinous price, and the arrangement must be condemned altogether.

A more moderate suppression of the central parts is, however, sometimes advantageous. Theory and experiment alike prove that a double line, of which the components are equally strong, is better resolved when, for example, one-sixth of the horizontal aperture is blocked off - by a central screen; or the rays quite at the centre may be allowed to pass, while others a little further removed are blocked off. Stops, each occupying one-eighth of the width, and with centres situated at the points of trisection, answer well the required purpose.

It has already been suggested that the principle of energy requires that the general expression for I 2 in (2) when integrated over the whole of the plane ξ, η should be equal to A, where A is the area of the aperture. A general analytical verification has been given by Stokes. The expression for I 2 may be written in the form 12= λ⅛ 2 fffJ ^ c ° s 7 j t(2' ’ ' - ®) + (3∕ ’- 2∕) J dxd l∕ d ^ d lf O)» the integrations with respect to x , , y , as well as those with respect to x, y being over the area of the aperture; and for the present purpose this is to be integrated again with respect to ξ, η over the whole of the focal plane.

In changing the order of integration so as to take first that with respect to ξ, -η, it is proper, in order to avoid ambiguity, to introduce under the integral sign the factor the + or -

being chosen so as to make the elements of the integral vanish at infinity. After the operations have been performed, α and β are to be supposed to vanish.

Thus J^fVdξdy = Limit of

' ψαiψ ^ cos 7 { i(χ, - χ ") + ^y , -y)}

τ√r ∕*+ x ^f^ α i 7 „ , 2a cos II

bow J„ e cos(Λξ-H) f Zξ=; and thus j∕∕"+∞ I 2 dξ <⅛ = Limit of

1 //TT" 4α3 dx dy dx' dy' WjJJ j α > + !ii^∣ S s = + ⅛→∑7 Let κ(x ai⅜ _ dx'='-du. J «

The limits for u are ultimately - ∞ and + ∞, and we have τ . ., Γ 2adx' 2f f+* du 2/ .

11111 J 1 2 ∣c 2 (¾ , - ¾) a ) κj. a l+w 2 κ ' π

⅛ ∙ + -jw-1

In like manner the integration for f may be performed; and we find y7"j^ 0 Γ I 2 dξdη = ffdxdy=A .... (10).

We saw that I 2 (the intensity at the focal point) was equal to K i ∣J∣Pf i . If A' be the area over which the intensity must be Iθ in order to give the actual total intensity in accordance with ΑΤ Ο 2 =#_1”Ι 2 ^^ > the relation between A and A' is AA'=λ 2 ∕ 2 . Since A' is in some sense the area of the diffraction pattern, it may be considered to be a rough criterion of the definition, and we infer that the definition of a point depends principally upon the area of the aperture, and only in a very secondary degree upon the shape when the area is maintained constant.

§ 12. Theory of Circular Aperture.

We will now consider the important ease where the form of the aperture is circular. Writing for brevity κ Cf=Ρ, κp∣f=g, (1), we have for the general expression (§ ll) of the intensity

λ 2 ∕ 2 I 2 -S 2 + C a (2), where S = JJsw,(px + qf) dxdy, (3),

C = JJ cos (px + qy) dxdy, (4).

When, as in the application to rectangular or circular apertures, the form is symmetrical with respect to the axes both of x and y, S = 0, and C reduces to

C= ff cos px cos qy dx dy, . .... (5).

In the case of the circular aperture the distribution of light is of course symmetrical with respect to the focal point p = 0, q≈0; and C is a function of p and q only through √(p 2 + ⅛ 2 ). It is thus sufficient to determine the intensity along the axis of ρ. Putting q - 0, we get

C = fffospxdxdy=2 f^cοs>ρx∖f(M 2 -x")dx,

R being the radius of the aperture. This integral is the Bessel’s function of order unity, defined by

J 1 (2) = -∕ cos (a cos ≠) sin 2 <∕> dφ . . . . (6).

7r √0

Thus, if x=Rcos≠, and the illumination at distance r from the focal point is 2 ∕2wRr∖

i2=r β4 1 ∖ A ) ,< n λM 2 ‘ ∕2ττRr∖2 k λ

∖ A 7

The ascending series for J 1 (z), used by Airy in his original investigation of the diffraction of a circular object-glass, and readily obtained from (6), is 3 2 s s s c 7

j i(s ) = 2 - 2 2 ^4 + 2 2 .4 2 T^6~2 2 . 4 2 .6 2 . 8 + * , ,

When z is great, we may employ the semi-convergent series τ , λ ∕∕ 2 ∖ . , 1 J 1 3.5.1∕1∖ 2 j >< l > - √⅛) 5in <≈- w i 1+ τπβD ¾.5.7,S>.l.S.5∕ iγ I

8.16. 24. 32 ∖zj ‘ $ K 2 ∖ ∕ 1 X ί 3 1 3.5.7.1.3/1V

+ √⅛) c0s < 2 -^i 8 ∙7~ 8.16.24 ∖z) 3.5.7.9,11.1.3.5.7 ∕iγ ∖

8. 16.24.32.40 L ) ' ΠΌ

Α table of the values of 2a^ 1 J 1 (a) has been given by Lommel, to whom is due the first systematic application of Bessel’s functions to the diffraction integrals.

The illumination vanishes in correspondence with the roots of the equation J 1 (≡) = 0. If these be called z 1 , z 2 , z 3 , . . . the radii of the dark rings in the diffraction pattern are fh≈ι A⅞ 2πR ’ 2πR ’ being thus inversely proportional to R.

The integrations may also be effected by means of polar coordinates, taking first the integration with respect to φ so as to obtain the result for an infinitely thin annular aperture. Thus, if x=ρcοsφ, y=psinφ,

C = ffeοs pxdxdy≈ ∕J'ff π cos (pp cos θ) pdp dθ.

Now by definition

J°(z) = ∣j∕*" ,rco s(= co s0)dθ = 1 -^ + 22-42 - 2 2 Γ4 2 76 2+ ' ' * The value of C for an annular aperture of radius r and width dr is thus d(J = 2πJ 0 (pρ)ρ dp, (12).

For the complete circle,

,, 2π∙ ∕~P^∙ τ f ∖ j C — ~p J J o(z ) 3 -⅛γ)⅛ 2 -^ r4 + ? /R6 ( - p 2 ⅛ 2 2^ a 74 + 2 2 . 4 2 . 6 ∙ ’ ∙∫

no 2J 1 (p r ) =1rl ‘ 7hΓ ’ as before.

In these expressions we are to replace p by klff, or rather, since the diffraction pattern is symmetrical, by krlf, where r is the distance of any point in the focal plane from the centre of the system.

The roots of J 0 (z) after the first may be found from z . ng ∙050661 ∙053041 , ∙262051 , ιsn

— = i — · 25 Ί -τ~. = .-τ~. rτπ + r~r-. ∑rτ⅞ · · k 1 ό)> π 4t-l (4t-l) j (4i-l) 5 and those of J 1 (z) from z . n , ’ 151982 ∙015399 ‘245835 ....

i-* + ' 2 °-w+Γ + 3W'(4iVi) i ’. ' ( ’’ formulae derived by Stokes from the descending series. The following table gives the actual values:— [table]

In both cases the image of a mathematical point is thus a symmetrical ring system. The greatest brightness is at the centre, where dC = 2πρ dp , C = πR 2 .

For a certain distance outwards this remains sensibly unimpaired, and then gradually diminishes to zero, as the secondary waves become discrepant in phase. The subsequent revivals of brightness forming the bright rings are necessarily of inferior brilliancy as compared with the central disk.

The first dark ring in the diffraction pattern of the complete circular aperture occurs when r∕∕= l∙2197×λ∕2R (15).

We may compare this· with the corresponding result for a rectangular aperture of width a, <// = a/«; and it appears that in consequence of the preponderance of the central parts, the compensation in the case of the circle does not set in at so small an obliquity as when the circle is replaced by a rectangular aperture, whose side is equal to the diameter of the circle.

Again, if we compare the complete circle with a narrow annular aperture of the same radius, we see that in the latter case the first dark ring occurs at a much smaller obliquity, viz., r∣f = - 7655 × λ∕2R.

It has been found by Herschel and others that the definition of a telescope is often improved by stopping off a part of the central area of the object-glass; but the advantage to be obtained in this way is in no case great, and anything like a reduction of the aperture to a narrow annulus is attended by a development of the total illumination distributed over the area of the circle of radius r. external luminous rings sufficient to outweigh any improvement due to the diminished diameter of the central area.

The maximum brightnesses and the places at which they occur are easily determined with the aid of certain properties of the Bessel’s functions. It is known that j 0 ’ ω=-j 1 ω, (iß);

J 2 (≈)=f J i (≈) - J√(≈) (17);

Jo(≡)+J a ω=∣J 1 ω (is)·

The maxima of C occur when d∕ j ι(z)∖ J 1 '(s) J 1 (s) dz∖ z ) z z 2 or by (17) when J a (s) = 0. When z has one of the values thus determined,

— J 1 (z) = J 0 (z).

The accompanying table is given by Lommel, in which the first column gives the roots of J 2 (z) = 0, and the second and third columns the corresponding values of the functions specified. It appears that the maximum brightness in the first ring is only about - ε 1 7 of the brightness at the centre.

We will now investigate the [table]

We have

4J⅛)

l λ y 2 · 2 δ l lj Λ

where z = 2πRr∕λ∕ (20).

Thus

2 tryI 2 rdr = ^^ J I⅜dz = 1rR 2 . 2 J~z~'s 2 tf)dz.

Now by (17), (18)

2 ’ 1 Jι(2 ) = J<∕ 2 ) — J∕(2 ); so that 2 ^ 1 J 1 2 ( 2 )=-⅛⅛Jo 2 ( 2 )-⅛⅛J 1 2 ( 2 ), aud

2y^ z s^ 1 J 1 2 (s) ds=l - J o 2 (2 ) - J 1 2 (s) . . . . (21).

If r, or z, be infinite, J 0 (z), J 1 (z) vanish, and the whole illumination is expressed by πR 2 , in accordance with the general principle. In any case the proportion of the whole illumination to be found outside the circle of radius r is given by

J 0 2 (z) + J 1 2 (z).

For the dark rings J 1 (z) = 0j so that the fraction of illumination outside any dark ring is simply J 0 2 (z). Thus for the first, second, third, and fourth dark rings we get respectively ∙161, ’ 090, ‘062, Ό47, showing that more than τ ⅞ths of the whole light is concentrated within the area of the second dark ring.

When z is great, the descending series (10) gives ^ i τ∣√(≡)' to < r -'W · · · < 22 >i so that the places of maxima and minima occur at equal intervals.

The mean brightness varies as z~ 3 (or as r^ 3 ), and the integral found by multiplying it by zdz and integrating between 0 and ∞ converges.

It may be instructive to contrast this with the case of an infinitely narrow annular aperture, where the brightness is proportional to J 0 2 (z). When z is great,

J o (2 ) = /y/(⅛) co ≡(s ≈-i ,r )∙

The mean brightness varies as« -1 ; and the integral^ x J 0 2 (z)zdz is not convergent.

The efficiency of a telescope is of course intimately connected with the size of the disk by which it represents a mathematical point. The resolving power upon double stars of telescopes of various apertures has been investigated by Dawes and others (Optics, vol. xvii. p. 807), with results that agree fairly well with theory.

If we integrate the expression (8) for I 2 with respect to r∖, we shall obtain a result applicable to a linear luminous source of which the various parts are supposed to act independently.

From (19), (20)

d ff v < h > ~ 2 ff d if ” fe - 2j Λ≈> ⅛- 2 ≡ ’ √⅞~ since η 2 = r 2 -ξ 2 .

If we write f = 2πRξ∕λ∕ (23), weg» 1 . . (24 ).

This integral has been investigated by H. Struve, who, calling to his aid various properties of Bessel’s functions, shows that

∕' co J 1 2 (z)<⅛ 2 1 ∕*i" ’. , .

Λ z√(z 2 -{∙ 2 j≡√ 7¾ sιn (2^sιn0) cos 2 ∕3 tf∕3 . (25), of which the right-hand member is readily expanded in powers of f. By means of (24) we may verify that

Z+∞ ∕*+∞

dξ ∕ I 2 dη = 7rR 2 .

∞ J— 00

Contrary to what would naturally be expected, the subject is more easily treated without using the results of the integration with respect to x and y, by taking first of all, as in the investigation of Stokes (§11), the integration with respect to η. Thus À 2 / 2 ∕" +co J¾ j = Limit of

Fβη -∕-∞

[[∕Jf β cos γ^ξ( x l- χ ) + r)(.f~y)]dxdydx'dfdη. (26); an( l J^-∞ e ^^ cos f j ξ(f- χ ) + v(f~y y ) dv

2/3 cos y (af - ae)

(27) ·

P -ty 2 We have now to consider ff 2β dy dy , ff β 2 + κ∖f

In the integration with respect to f every element vanishes in the limit (3 = 0), unless y , =y. If the range of integration for f includes the value y, then Limit j⅞+W⅛Z7 3=a; otherwise it vanishes.

The limit of (28) may thus be denoted by λ∕ Y, where Y is the common part of the ranges of integration for if and y corresponding to any values of x ’ and x. Hence f-∞ = λ ’ 1 ∕^ γ JJ~ Y cos y (aj' - at) dx dx! cοs^γ-eοs d χ dx! · · (29), if, as for the present purpose, the aperture is symmetrical with respect to the axis of y.

In the application to the circle we may tvrite

+ c ° I 2 ⅜ = 4λ - 1 ∕- R Y cos cos^ dx dx , , where Y is the smaller of the two quantities

2√(R 2 -a∕ 2 ), 2√(R 2 -x 2 ), ue., corresponds to the larger of the two abscissae af, x. If we take Y = 2√(R 2 -z 2 ), and limit the integration to those values of x , which are less than x, we should obtain exactly the half of the required result. Thus ∕" + I 2 dη = 16 λ ^ 1 ∕ -1 f f √(R 2 -ic 2 )cos^'cos^i darife'

√-∞ √o √o f f = √(R 2 - a? 2 ) sin cfø

7r <√o. ∕

Hence, writing as before f= , we gθ∣,

∕'*4 - ∞ 4R 2 f7Λ ∕~⅛7r f ⅛J χ = VÎT ‘ ^fj∕ 0 cos 2 βsin(2(jg)d0. . (30) j in which we may replace dξ∕ξ by dζ∕ζ, in agreement with the result obtained by Struve.

The integral in (30) may be written in another form. We have ∕ < b sin(2^3)cos 2 3√3

= -⅜cos3cos(2< u sin3)-⅛ycos(2<sin∕3)sin∕3 dβ; and thus ζsin(2ζβ)cοs 2 βdβ = ⅛^ j 1 - cos(2<sin0) I sin βdβ = sin 2 (<sin3)sin3√3 (31).

The integral is thus expressible by means of the function K 1 , and we have

^^I 2 ^ = ⅜R 2 <- 3 d^K 1 (20 . . . (32).

The ascending series for K 1 (z) is κ ∙<≈HHi⅛-id^i≡3⅛rr∙∙∙!; (33); and this is always convergent. The descending semi-convergent series is

K 1 (z) = ^ j z+z^ 1 -3.z^ 8 + l 2 .3 2 .5.z^ 5 -l a .3 2 .5 2 .7.z^ 7 + . . . ∣

- √(∣)∙-<≈-w1 1 -⅛⅛⅛-+ ·∙∙· I -√(⅞)∙^-^l⅛ 4 - tl, ^¾^-∣^ the series within braces being the same as those which occur in the expression of the function J 1 (z). When C (or ξ) is very great, c lξ ∕ +α ⅛=-r 2 <. k ∕-00 7Γ

so that the intensity of the image of a luminous line is ultimately inversely as the square of the distance from the central axis, or geometrical image.

As is evident from its composition, the intensity remains finite for all values of C; it is, however, subject to fluctuations presenting maxima and minima, which have been calculated by Ch. André, using apparently the method of quadratures.

The results are also exhibited by Μ. André in the form of a curve, of which fig. 6 is a copy. [table]

It will be seen that the distribution ot brightness does not differ greatly from that due to a rectangular aperture whose width (perpendicular to the luminous line) is equal to the diameter of the circular aperture. It will be instructive to examine the image of a double line, whose components present an interval corresponding to ζ=π, and to compare the result with that already found for a rectangular aperture (§ ll). We may consider the brightness at distance £ proportional to

τ f ∩ 1 2 ^ + 24 < 4 — · · · f35 l mu- 1 2 .3 1 2 .3 2 .5 + 1 2 .3 2 .5 2 .7 ' , '

In the compound image the illumination at the geometrical focus of one of the luminous lines is represented by

L(0) + L(π)j and the illumination midway between the geometrical images of the two lines is

2L(⅛π),

We find by actual calculation from the series, L(π)=∙0164, L(⅜π) = *1671, L(0) = ’ 3333, so that

L(O) + L(ιr) = -3497, 2L(⅜π)='3342, and 2L(⅜ιr) _

L(0) + L(π) - 9δ5 '

The corresponding number for the rectangular aperture was - 811; so that, as might have been expected, the resolving power of the circular aperture is distinctly less than that of the rectangular aperture of equal width. Hence a telescope will not resolve a double line unless the angular interval between them decidedly exceeds that subtended by the wave-length of light at a distance equal to the diameter of the object-glass. Experiment shows that resolution begins when the angular interval is about a tenth part greater than that mentioned.

If we integrate (30) with respect to ξ between the limits - ∞ and + oo we obtain πR 2 , as has already been remarked. This represents the whole illumination over the focal plane due to a radiant point whose image is at 0, or, reciprocally, the illumination at 0 (the same as at any other point) due to an infinitely extended luminous area. If we take the integration from ξ (supposed positive) to ∞ we get the illumination at 0 due to a uniform luminous area extending over this region, that is to say, the illumination at a point situated at distance ξ outside the border of the geometrical image of a large uniform area. If the point is supposed to be inside the geometrical image and at a distance £ from its edge, we are to take the in tegration from - ∞ to ξ. Thus, if we choose the scale of intensities so that the full intensity is unity, then the intensity at a distance corresponding to + ζ (outside the geometrical image) may be represented by E ( + ζ), and that at a distance - £ by E( - ζ), where

E( + O+3E(-0-l, and

E(0 = Ty<V 3 ^K 1 (2<) = ⅛-l^- 3 ^K 1 (2θ. . (36).

This is the result obtained by Struve, who gives the following series for E(£).

The ascending series, obtained at once by integration from (33), is TO-⅛- ? ¾ ι -υ- 1 ⅞⅛ 1 ,, y √.. (2n÷iy · . < w >∙

When C is great, we have approximately from the descending series εil .Vl + J-Ι- t "∣⅜÷w λ ' √A( 12f a √ 2i s ' a C' ,

Thus “at great distances from the edge of the geometrical image the intensity is inversely proportional to the distance, and to the radius of the object-glass.”

The following table, abbreviated from that given by Struve, will serve to calculate the enlargement of an image due to diffraction in any case that may arise.

C=2πRξ∕λ∕.

3E(-C≈l-≡ + 0. [table]

It may perhaps have struck the reader that there is some want of rigour in our treatment of (30) when we integrate it over the whole focal plane of ξ, -η, inasmuch as in the proof of the formulae ξ and η are supposed to be small. The inconsistency becomes very-apparent when we observe that according to the formulae there is no limit to the relative retardation of secondary waves coming from various parts of the aperture, whereas in reality this retardation could never exceed the longest line capable of being drawn within the aperture. It will be worth while to consider this point a little further, although our limits forbid an extended treatment. _

The formula becomes rigorous if we regard it as giving the illumination on the surface of a sphere of very large radius ∕, in a direction such that

ξ=∕sin0cos≠, η=∕sin0sinφ; it may then be written p=A-2y-2^//27*cos κ I (af-oe)sin θ cos φ + (y'-y)sin Osin φ}dxdydx'df.

The whole intensity over the infinite hemisphere is given by E=√ 2 ∕^ ’ 7"^T 2 sin0dθd≠ (38).

√o √o

According to the plan formerly adopted, we postpone the integration with respect to x, y, x ’ , y ’ , and take first that with respect to θ and φ. Thus for a single pair of elements of area dxdy, dx!dy ’ we have to consider JJ" cosκ{(z' -ic)sin0cos≠ + (y -y} sin0sinφ ∙-sin0 dθ dφ, or, if we write x ’-≈=rcosα, ?/-y = rsina,

∕*⅜π∙ ∕*2ιr

∕ ∕ cos(κrsin0cos≠)sin0 dθ dφ.

√0 √0

Now it may be proved {e.g., by expansion in powers of κr) that ∕*⅛π∙∕*2π∙ . sinκr ∕ ∕ cos{κrsinθcosφ')sinθ dθ dφ = 2π ——- . . (39);

√o √o κ '

and thus ≡ “JJJf dxdydx ’ df . . . . (40), r being the distance between the two elements of area dx dy, dx'df.

In the case of a circular area of radius R, we have ∕77∕' sinκr . 7 , , 7 . 2ιrβ2 ί ί Jι(2κR) )

JJJJ ~^V dxd y dxd y = ∖ υ ~- κ ^τ∖' and thus

E = πR 2 j I"— (41 >'

When κR = oo,

E = πR 2 , as before.

It appears therefore that according to the assumed law of the secondary wave the total illumination is proportional to the area of aperture, only under the restriction that the linear dimensions of the aperture are very large in comparison with the wave-length.

A word as to the significance of (39) may not be out of place. We know that

Ψ = cosκ{sin0cos<∕>x + sin0sin≠y + cos0s} . . . (42)

satisfies Laplace’s extended equation (v 2 + κ 2 )ψ = 0, being of the form cosκx', where x ’ is drawn in an oblique direction; and it follows that ffψsinθdθdψ satisfies the same equation. Now this, if the integration be taken over the hemisphere 0 = 0 to 0 = ⅛π, must become a function of r, or ∖∕(≈ 2 + y 2 + ≡ 2 ), only.

Hence, putting x =r, y = 0, z = 0, we get

/V* ∕* 3 7r ∕*^ 7γ

∕∕ψsin0fZ0iZ≠=y J cos (κr sin 0 cos <∕>) sin 0iZ0iZ≠.

But the only function of r which satisfies Laplace’s equation continuously through the origin is A sin κr∕(κr) ; and that A=2π is proved at once by putting r = 0. The truth of the formula may also be established independently of the differential equation by ∕*g7Γ ∕*2ιr equating the values ofyf J ψ sin 0 dθdφ, when x=r, y = 0, z≈Q, and when x = Q, y=0, z≈∙r. Thus

∕**⅛ιr ∕ - 27Γ .

∕ ∕ cos (κr sin 0 cos ≠) sin θdθdφ J 0 -∕o ∕*5 7γ ∕**- 7γ sin κP =J^ cos (κrcosθ)sin θdθdφ = 2π·

The formula itself may also be written

∕* J, τr sin κτ Je J o (κrsin0)sin0d0 = -^- .... (43).

The results of the preceding theory of circular apertures admit of an interesting application to coronas, such as are often seen encircling the sun and moon. They are due to the interposition of small spherules of water, which act the part of diffracting obstacles. In order to the formation of a well-defined corona it is essential that the particles be exclusively, or preponderatingly, of one size.

If the origin of light be treated as infinitely small, and be seen in focus, whether with the naked eye or with the aid of a telescope, the whole of the light in the absence of obstacles would be concentrated in the immediate neighbourhood of the focus. At other parts of the field the effect is the same, by Babinet’s principle, whether the imaginary screen in front of the object-glass is generally transparent but studded with a number of opaque circular disks, or is generally opaque but perforated with corresponding apertures. Consider now the light diffracted in a direction many times more oblique than any with which we should be concerned, were the whole aperture uninterrupted, and take first the effect of a single small aperture. The light in the proposed direction is that determined by the size of the small aperture in accordance with the laws already investigated, and its phase depends upon the position of the aperture. If we take a direction such that the light (of given wave-length) from a single aperture vanishes, the evanescence continues even when the whole series of apertures is brought into contemplation. Hence, whatever else may happen, there must be a system of dark rings formed, the same as from a single small aperture. In directions other than these it is a more delicate question how the partial effects should be compounded. If we make the extreme suppositions of an infinitely small source and absolutely homogeneous light, there is no escape from the conclusion that the light in a definite direction is arbitrary, that is, dependent upon the chance distribution of apertures. If, however, as in practice, the light be heterogeneous, the source of finite area, the obstacles in motion, and the discrimination of different directions imperfect, we are concerned merely with the mean brightness found by varying the arbitrary phase-relations, and this is obtained by simply multiplying the brightness due to a single aperture by the number of apertures (n). The diffraction pattern is therefore that due to a single aperture, merely brightened n times.

In his experiments upon this subject Fraunhofer employed plates of glass dusted over with lycopodium, or studded with small metallic disks of uniform size; and he found that the diameters of the rings were proportional to the length of the waves and inversely as the diameter of the disks.

In another respect the observations of Fraunhofer appear at first sight to be in disaccord with theory; for his measures of the diameters of the red rings, visible when white light was employed, correspond with the law applicable to dark rings, and not to the different law applicable to the luminous maxima. Verdet has, however, pointed out that the observation in this form is essentially different from that in which homogeneous red light is employed, and that tho position of the red rings would correspond to the absence of blue-green light rather than to the greatest abundance of red light. Verdet’s own observations, conducted with great care, fully confirm this view, and exhibit a complete agreement with theorv.

By measurements of coronas it is possible to infer the size of the particles to which they are due, an application of considerable interest in the case of natural corona—the general rule being the larger the corona the smaller the water spherules. Young employed this method not only to determine the diameters of cloud particles (e.g., χ-∏oτy inch), but also those of fibrous material, for which the theory is analogous. His instrument was called the eriometer.

§ 13. Influence of Aberration. Optical Power of Instruments.

Our investigations and estimates of resolving power have thus far proceeded upon the supposition that there are no optical imperfections, whether of the nature of a regular aberration or dependent upon irregularities of material and workmanship. In practice there will always be a certain aberration or error of phase, which we may also regard as the deviation of the actual wavesurface from its intended position. In general, we may say that aberration is unimportant, when it nowhere (or at any rate over a relatively small area only) exceeds a small fraction of the wavelength (λ). Thus in estimating the intensity at a focal point, where, in the absence of aberration, all the secondary waves would have exactly the same phase, we see that an aberration nowhere exceeding ⅛λ can have but little effect.

The only case in which the influence of small aberration upon the entire image has been calculated is that of a rectangular aperture, traversed by a cylindrical wave with aberration equal to cx i . The aberration is here unsymmetrical, the wave being in advance of its proper place in one half of the aperture, but behind in the other half. No terms in x or x 2 need be considered. The first would correspond to a general turning of the beam; and the second would imply imperfect focusing of the central parts. The effect of aberration may be considered in two ways. We may suppose the aperture (a) constant, and inquire into the operation of an increasing aberration; or we may take a given value of c (i.e., a given wave-surface) and examine the effect of a varying aperture. The results in the second case show that an increase of aperture up to that corresponding to an extreme aberration of half a period has no ill effect upon the central band (§ 11), but it increases unduly the intensity of one of the neighbouring lateral bands; and the practical conclusion is that the best results will be obtained from an aperture giving an extreme aberration of from a quarter to half a period, and that with an increased aperture aberration is not so much a direct cause of deterioration as an obstacle to the attainment of that improved definition which should accompany the increase of aperture.

If, on the other hand, we suppose the aperture given, we find that aberration begins to be distinctly mischievous when it amounts to about a quarter period, i.e., when the wave-surface deviates at each end by a quarter wave-length from the true plane.

For the focal point itself the calculations are much simpler. We will consider the case of a circular object-glass with a symmetrical aberration proportional to hρ 4 . The vibration will be represented by

2 j∕^ 1 cos (nt - hρ i ) ρ dρ, in which the radius of the aperture is supposed to be unity. The intensity is thus expressed by

U= ^2 j ^' 1 cqs(7tp 4 )ptZpJ + ^2^ 1 sin(Ap 4 )prfpJ . . . (1), the scale being such that the intensity is unity when there is no aberration (h = 0).

By integration by parts it can be shown that

√0 p p ί 6 6.10 6.10.14 + ί ’

so that

2^∙ 1 ∞ s (V)p⅛-∞,⅛j 1 -0.⅛ s + -^A-- ... j +sto M⅛-βS + ∙∙∙i · · · · (2) · 2 ∕^ 1 sin(∕φ 4 )p dp = sinΛ j 1- W + O* . . . j

√ 0 κ ( 6.10 6.10.14.18 ί

- cos √^-W- + ..Λ (3).

I 6 6.10.14 ∕ k '

Hence, when h≈⅛π,

2 cos (iπp 4 ) P dp = 1 ∙32945∕√2,

2 j 4' 1 sin(⅜πp 4 )p i Zp= ∙35424∕√2,

I< 2 ,= ∙9464.

Similarly, when h=½π,

Iθ= ’ 8003; and when h = π,

I>∙3947.

These numbers represent the influence of aberration upon the intensity at the central point, upon the understanding that the focusing is that adapted to a small aperture, for which h might be neglected. If a readjustment of focus be permitted, the numbers will be sensibly raised. The general conclusion is that an aberration between the centre and circumference of a quarter period has but little effect upon the intensity at the central point of the image.

As an application of this result, let us investigate what amount of temperature disturbance in the tube of a telescope may be expected to impair definition. According to Biot and Arago, the index μ for air at t° C. and at atmospheric pressure is given by •00029

μ 1- l + ∙0037U

If we take 0° C. as standard temperature, δ j a = -Γli×10- 6 .

Thus, on the supposition that the irregularity of temperature t extends through a length l, and produces an acceleration of a quarter of a wave-length, jλ=l ’ l Zi × 10^ β ; or, if we take λ = 5 ’ 3×10^ 5 ,

Zi—12, the unit of length being the centimetre.

We may infer that, in the case of a telescope tube 12 cm. long, a stratum of air heated 1° C. lying along the top of the tube, and occupying a moderate fraction of the whole volume, would produce a not insensible effect. If the change of temperature progressed uniformly from one side to the other, the result would be a lateral displacement of the image without loss of definition; but in general both effects would be observable. In longer tubes a similar disturbance would be caused by a proportionally less difference of temperature.

We will now consider the application of the principle to the formation of images, unassisted by reflexion or refraction. The f unction of a lens in forming an image is to compensate by its variable thickness the differences of phase which would otherwise exist between secondary waves arriving at the focal point from various parts of the aperture (Optics, vol. xvii. p. 802). If we suppose the diameter of the lens to be given (2R), and its focal length f gradually to increase, the original differences of phase at the image of an infinitely distant luminous point diminish without limit. When f attains a certain value, say f 1 . the extreme error of phase to be compensated falls to ¼λ. But, as we have seen, such an error of phase causes no sensible deterioration in the definition; so that from this point onwards the lens is useless, as only improving an image already sensibly as perfect as the aperture admits of. Throughout the operation of increasing the focal length, the resolving power of the instrument, which depends only upon the aperture, remains unchanged; and we thus arrive at the rather startling conclusion that a telescope of any degree of resolving power might be constructed without an object-glass, if only there were no limit to the admissible focal length. This last proviso, however, as we shall see, takes away almost all practical importance from the proposition.

To get an idea of the magnitudes of the quantities involved, let us take the case of an aperture of ⅕ inch, about that of the pupil of the eye. The distance f 1 , which the actual focal length must exceed, is given by

√(∕ι 2 + R 2 )-∕ι = ⅛λ; so that

∕ 1 = 2R-∕λ (4).

Thus, if λ = 1 / 40000 R= 1 / 10 we find f 1 = 800 inches.

The image of the sun thrown upon a screen at a distance exceeding 66 feet, through a hole 1 / 5 inch in diameter, is therefore at least as well defined as that seen direct.

As the minimum focal length increases with the square of the aperture, a quite impracticable distance would be required to rival the resolving power of a modern telescope. Even for an aperture of 4 inches, f 1 would have to be 5 miles.

A similar argument may be applied to find at what point an achromatic lens becomes sensibly superior to a single one. The question is whether, when the adjustment of focus is correct for the central rays of the spectrum, the error of phase for the most extreme rays (which it is necessary to consider) amounts to a quarter of a wave-length. If not, the substitution of an achromatic lens will be of no advantage. Calculation shows that, if the aperture be ⅛ inch, an achromatic lens has no sensible advantage if the focal length be greater than about 11 inches. If we suppose the focal length to be 66 feet, a single lens is practically perfect up to an aperture of 1∙7 inch.

Some estimates of the admissible aberration in a spherical lens have already been given under Optics, vol. xvii. p. 807. In a similar manner we may estimate the least visible displacement of the eye-piece of a telescope focused upon a distant object, a question of interest in connexion with range-finders. It appears that a displacement δ f from the true focus will not sensibly impair definition, provided

δ∕<∕ 2 λ∕R 2 , (5),

2R being the diameter of aperture. The linear accuracy required is thus a function of the ratio of aperture to focal length. The formula agrees well with experiment.

The principle gives an instantaneous solution of the question of the ultimate optical efficiency in the method of “mirror-reading,” as commonly practised in various physical observations. A rotation by which one edge of the mirror advances ∣λ (while the other edge retreats to a like amount) introduces a phase-discrepancy of a whole period where before the rotation there was complete agreement. A rotation of this amount should therefore be easily visible, but the limits of resolving power are being approached; and the conclusion is independent of the focal length of the mirror, aud of the employment of a télescope, provided of course that the reflected image is seen in focus, and that the full width of the mirror is utilized.

A comparison with the method of a material pointer, attached to the parts whose rotation is under observation, and viewed through a microscope, is of interest. The limiting efficiency of the microscope is attained when the angular aperture amounts to 180° (Microscope, vol. xvi. p. 267; Optics, vol. xvii. p. 807); and it is evident that a lateral displacement of the point under observation through ½λ entails (at the old image) a phase-discrepancy of a whole period, one extreme ray being accelerated and the other retarded by half that amount. We may infer that the limits of efficiency in the two methods are the same when the length of the pointer is equal to the width of the mirror.

An important practical question is the amount of error admissible in optical surfaces. In the case of a mirror, reflecting at nearly perpendicular incidence, there should be no deviation from truth (over any appreciable area) of more than ⅜λ. For glass, μ - 1=½ nearly; and hence the admissible error in a refracting surface of that material is four times as great.

In the case of oblique reflexion at an angle φ , the error of retardation due to an elevation BD (fig. 7) is

QQ ’- QS = BDsecφ(l - cosSQQ ’) = BDsec≠(l + cos2≠) = 2BDcos≠; from which it follows that an error of given magnitude in the figure of a surface is less important in oblique than in perpendicular reflexion. It must, however, be borne in mind that errors can sometimes be compensated by altering adjustments. If a surface intended to be flat is affected with a slight general curvature, a remedy may be found in an alteration of focus, aud the remedy is the less complete as the reflexion is more oblique.

The formula expressing the optical power of prismatic spectroscopes is given with examples under Optics, vol. xvii. p. 807, and may readily be investigated upon the principles of the wave theory. Let A 0 B 0 (fig. 8) be a plane wave-surface of the light before it falls upon the prisms, AB the corresponding wave-surface for a particular part of the spectrum after the light has passed the prisms, or after it has passed the eye-piece of the observing telescope. The path of a ray from the wave-surface A 0 B 0 to A or B is determined by the condition that the optical distance, fμ ds, is a minimum (Optics, vol. xvii. p. 798); and, as AB is by supposition a wave-surface, this optical distance is the same for both points. Thus Jμds(jοτ A) =j^μds (for B) (6).

We have now to consider the behaviour of light belonging to a neighbouring part of the spectrum. The path of a ray from the wave-surface A 0 B 0 to the point A is changed; but in virtue of the minimum property the change may be neglected in calculating the optical distance, as it influences the result by quantities of the second order only in the changes of refrangibility. Accordingly, the optical distance from A 0 B 0 to A is represented by f (μ + δμ)rfs, the integration being along the original path A o ... A; and similarly the optical distance between A 0 B 0 and B is represented by y (ju +δ j α)ιfs, the integration being along B 0 . . . B. In virtue of (6) the difference of the optical distances to A and B is y δμds (along B o . . . B) - y bμds (along A o . . . A) . (7).

The new wave-surface is formed in such a position that the optical distance is constant; and therefore the dispersion, or the angle through which the wave surface is turned by the change of refrangibility, is found simply by dividing (7) by the distance AB. If, as in common flint-glass spectroscopes, there is only one dispersing substance, fbμ ds=δμ.s, where s is simply the thickness traversed by the ray. If Z 2 and t 1 be the thicknesses traversed by the extreme rays, and a denote the width of the emergent beam, the dispersion θ is given by

0-δμ(f 2 -i 1 )∕α, or, if t 1 be negligible, θ-bμt∣a (8).

The condition of resolution of a double line whose components subtend an angle θ is that θ must exceed λ∕α. Hence, in order that a double line may be resolved whose components have indices μ and μ + Sμ, it is necessary that t should exceed the value given by the following equation :—

t=λ∣bμ (9).

For applications of these results, see Spectroscope.

§ 14. Theory of Gratings.

The general explanation of the mode of action of gratings has been given under Light (vol. xiv. p. 607). If the grating be composed of alternate transparent and opaque parts, the question may be treated by means of the general integrals (§11) by merely limiting the integration to the transparent parts of the aperture. For an investigation upon these lines the reader is referred to Airy’s Tracts and to Verdet’s Leçons. If, however, we assume the theory of a simple rectangular aperture (§ ll), the results of the ruling can be inferred by elementary methods, which are perhaps more instructive.

Apart from the ruling, we know that the image of a mathematical line will be a series of narrow bands, of which the central one is by far the brightest. At the middle of this band there is complete agreement of phase among the secondary waves. The dark lines which separate the bands are the places at which the phases of the secondary wave range over an integral number of periods. If now we suppose the aperture AB to be covered by a great number of opaque strips or bars of width cl, separated by transparent intervals of width a, the condition of things in the directions just spoken of is not materially changed. At the central point there is still complete agreement of phase; but the amplitude is diminished in the ratio of a: a + d. In another direction, making a srnall angle with the last, such that the projection of AB upon it amounts to a few wave-lengths, it is easy to see that the mode of interference is the same as if there were no ruling. For example, when the direction is such that the projection of AB upon it amounts to one wave-length, the elementary components neutralize one another, because their phases are distributed symmetrically, though discontinuously, round the entire period. The only effect of the ruling is to diminish the amplitude in the ratio a: a + d; and, except for tho difference in illumination, the appearance of a line of light is the same as if the aperture were perfectly free.

The lateral (spectral) images occur in such directions that the projection of the element (a + d) of the grating upon them is an exact multiple of λ. The effect of each of the n elements of the grating is then the same; and, unless this vanishes on account of a particular adjustment of the ratio a: d, the resultant amplitude becomes comparatively very great. These directions, in which the retardation between A and B is exactly mnκ, may be called the principal directions. On either side of any one of them the illumination is distributed according to the same law as for the central image (7n = O), vanishing, for example, when the retardation amounts to (≡±l)λ. In considering the relative brightnesses of the different spectra, it is therefore sufficient to attend merely to the principal directions, provided that the whole deviation be not so great that its cosine differs considerably from unity.

We have now to consider the amplitude due to a single element, which we may conveniently regard as composed of a transparent part a bounded by two opaque parts of width ⅛d. The phase of the resultant effect is by symmetry that of the component which comes from the middle of a. The fact that the other components have phases differing from this by amounts ranging between ±ατnπ∕(α + cQ causes the resultant amplitude to be less than for the central image (where there is complete phase agreement). If B m denote tho brightness of the wι th lateral image, and B o that of the central image, we have .

αrnπ

∏ . τ > Γ ∕^ a+t, i 2amτ~] 2 ha + d∖ 2 . .arnr ... B m :B 0 = ∕ cosa∙dz4 —- 1 = sin 2 3 . (1).

L√ - araπ a + d_] ∖ am* ∕ a + d ' ’ a+d

If B denote the brightness of the central image when the whole of the space occupied by the grating is transparent, we have

B o : B = α 2 =(α + d) 2 , and thus

τ> τ> 1 * 2 amTΓ ιa∖

B m : B = — 2 -⅛ sin - ——. (2).

wι⅛ 2 a + d

The sine of an angle can never be greater than unity; and consequently under the most favourable circumstances only l∕τn⅛ 2 of the original light can be obtained in the τn th spectrum. We conclude that, with a grating composed of transparent and opaque parts, the utmost light obtainable in any one spectrum is in the first, and there amounts to l∕ιr 2 , or about 1 ⅛, and that for this purpose α and d must be equal. When d=α, the general formula becomes

„ „ sin 2 ⅜ww

B,„:B=—⅛- (3), showing that, when m is even, B m vanishes, and that, when m is odd, B m : B = 1 ∕wι 2 ιr 2 .

The third spectrum has thus only ⅜ of the brilliancy of the first.

Another particular case of interest is obtained by supposing α small relatively to (α + d). Unless the spectrum be of very high order, we have simply

B,„: B= {α∕(α + d)} 2 (4); so that the brightnesses of all the spectra are the same.

The light stopped by the opaque parts of the grating, together with that distributed in the central image and lateral spectra, ought to make up the brightness that would be found in the central image, were all the apertures transparent. Thus, if α=d, we should have

, 1 1 , 2 ∕, 1 ,1 ∖

1 = 2 + 4 + √Λ 1 + 9 + 25 + ∙ , J’

which is true by a known theorem. In the general case a _ ί a ∖ 2 2 m ς,° 0 1 . 2 ∕ mιτa ∖

a + d ∖a + d) + π 2 n ,=ι «i aSln ∖α + d∕ ’ a formula which may be verified by Fourier’s theorem.

According to a general principle formulated by Babinet, the brightness of a lateral spectrum is not affected by an interchange of the transparent and opaque parts of the grating. The vibrations corresponding to the rivo parts are precisely antagonistic, since if both were operative the resultant would be zero. So far as the application to gratings is concerned, the same conclusion may be derived from (2).

From the value of B m : B 0 we see that no lateral spectrum can surpass the central image in brightness; but this result depends upon the hypothesis that the ruling acts by opacity, which is generally very far from being the case in practice. In an engraved glass grating there is no opaque material present by which light could be absorbed, and the effect depends upon a difference of retardation in passing the alternate parts. It is possible to prepare gratings which give a lateral spectrum brighter than the central image, and the explanation is easy. For if the alternate parts were equal and alike transparent, but so constituted as to give a relative retardation of ½λ, it is evident that the central image would be entirely extinguished, while the first spectrum would be four times as bright as if the alternate parts were opaque. If it were possible to introduce at every part of the aperture of the grating an arbitrary retardation, all the light might be concentrated in any desired spectrum. By supposing the retardation to vary uniformly and continuously we fall upon the case of an ordinary prism; but there is then no diffraction spectrum in the usual sense. To obtain such it would be necessary that the retardation should gradually alter by a wave-length in passing over any element of the grating, and then fall back to its previous value, thus springing suddenly over a wave-length. It is not likely that such a result will ever be fully attained in practice; but the case is worth stating, in order to show that there is no theoretical limit to the concentration of light of assigned wave-length in one spectrum, and as illustrating the frequently observed unsymmetrical character of the spectra on the two sides of the central image.

We have hitherto supposed that the light is incident perpendicularly upon the grating; but the theory is easily extended. If the incident rays make an angle θ with the normal (fig. 9), and the diffracted rays make an angle φ (upon the same side), the relative retardation from each element of width (a + d) to the next is (a + d) (sin θ +sin φ); and this is the quantity which is to be equated to m λ. Thus sin0 + sin<∕> = 2sin⅜(0 + ≠).cos∣(0 - φ)≈mλ∕(a + d) . (5).

The “deviation” is (θ+φ), and is therefore a minimum when 0 = ≠, i.e., when the grating is so situated that the angles of incidence and diffraction are equal.

In the case of a reflexion grating the same method applies. If θ and φ denote the angles with the normal made by the incident and diffracted rays, the formula (5) still holds, and, if the deviation be reckoned from the direction of the regularly reflected rays, it is expressed as before by (θ + φ), and is a minimum when θ = φ, that is, when the diffracted rays return upon the course of the incident rays.

In either case (as also with a prism) the position of minimum deviation leaves the width of the beam unaltered, i . e., neither magnifies nor diminishes the angular width of the object under view.

From (5) we see that, when the light falls perpendicularly upon a grating (θ = 0), there is no spectrum formed (the image corresponding to m = 0 not being counted as a spectrum), if the grating interval σ or (a + d ) is less than λ. Under these circumstances, if the material of the grating be completely transparent, the whole of the light must appear in the direct image, and the ruling is not perceptible. From the absence of spectra Fraunhofer argued that there must be a microscopic limit represented by λ; and the inference is plausible, to say the least. Fraunhofer should, however, have fixed the microscopic limit at ½λ, as appears from (5), when we suppose θ = ⅛τr, √> = ⅜ιr.

We will now consider the important subject of the resolving power of gratings, as dependent upon the number of lines (n) and the order of the spectrum observed (m ). Let BP (fig. 11) be the direction of the principal maximum (middle of central band) for the wave-length λ in the m th spectrum. Then the relative retardation of the extreme rays (corresponding to the edges A, B of the grating) is mnλ. If BQ be the direction for the first minimum (the darkness between the central and first lateral band), the relative retardation of the extreme rays is (mn + 1)λ. Suppose now that λ + δλ is the wave-length for which BQ gives the principal maximum, then

(mn + 1)λ=mn(λ + δλ); whence

δλ∕λ = 1 ∣mn . . (6).

According to our former standard, this gives the smallest difference of wave-lengths in a double line which can be just resolved; and we conclude that the resolving power of a grating depends only upon the total number of lines, and upon the order of the spectrum, without regard to any other considerations. It is here of course assumed that the n lines are really utilized.

In the case of the D-lines the value of δλ∕λ is about 1/1000; so that to resolve this double line in the first spectrum requires 1000 lines, in the second spectrum 500, aud so on.

It is especially to be noticed that the resolving power does not depend directly upon the closeness of the ruling. Let us take the case of a grating 1 inch broad, and containing 1000 lines, and consider the effect of interpolating an additional 1000 lines, so as to bisect the former intervals. There will be destruction by interference of the first, third, and odd spectra generally; while the advantage gained in the spectra of even order is not in dispersion, nor in resolving power, but simply in brilliancy, which is increased four times. If we now suppose half the grating cut away, so as to leave 1000 lines in half an inch, the dispersion will not be altered, while the brightness and resolving power are halved.

There is clearly no theoretical limit to the resolving power of gratings, even in spectra of given order. But it is possible that, as suggested by Rowland, the structure of natural spectra may be too coarse to give opportunity for resolving powers much higher than those now in use. However this may be, it would always be possible, with the aid of a grating of given resolving power, to construct artificially from white light mixtures of slightly different wave-length whose resolution or otherwise would discriminate between powers inferior and superior to the given one.

If we define as the “dispersion” in a particular part of the spectrum the ratio of the angular interval dθ to the corresponding increment of wave-length dλ, we may express it by a very simple formula. For the alteration of wave-length entails, at the two limits of a diffracted wave-front, a relative retardation equal to mndλ. Hence, if a be the width of the diffracted beam, and dθ the angle through which the wave-front is turned, adθ = mndλ, or dispersion = mn∣a (7).

The resolving power and the width of the emergent beam fix the optical character of the instrument. The latter element must eventually be decreased until less than the diameter of the pupil of the eye. Hence a wide beam demands treatment with further apparatus (usually a telescope) of high magnifying power.

In the above discussion it has been supposed that the ruling is accurate, and we have seen that by increase of m a high resolving power is attainable with a moderate number of lines. But this procedure (apart from the question of illumination) is open to the objection that it makes excessive demands upon accuracy. According to the principle already laid down, it can make but little difference in the principal direction corresponding to the first spectrum, provided each line lie within a quarter of an interval (a + d) from its theoretical position. But, to obtain an equally good result in the m th spectrum, the error must be less than 1/ m of the above amount.

There are certain errors of a systematic character which demand special consideration. The spacing is usually effected by means of a screw, to each revolution of which corresponds a large number (e.g., one hundred) of lines. In this way it may happen that, although there is almost perfect periodicity with each revolution of the screw after (say) 100 lines, yet the 100 lines themselves are not equally spaced. The “ghosts” thus arising were first described by Quincke, and have been elaborately investigated by Peirce, both theoretically and experimentally. The general nature of the effects to be expected in such a case may be made clear by means of an illustration already employed for another purpose. Suppose two similar and accurately ruled transparent gratings to be superposed in such a manner that the lines are parallel. If the one set of lines exactly bisect the intervals between the others, the grating interval is practically halved, and the previously existing spectra of odd order vanish. But a very slight relative displacement will cause the apparition of the odd spectra. In this case there is approximate periodicity in the half interval, but complete periodicity only after the whole interval. The advantage of approximate bisection lies in the superior brilliancy of the surviving spectra; but in any case the compound grating may be considered to be perfect in the longer interval, and the definition is as good as if the bisection were accurate.

The effect of a gradual increase in the interval (fig. 12) as we pass across the grating has been investigated by Cornu, who thus explains an anomaly observed by Mascart. The latter found that certain gratings exercised a converging power upon the spectra formed upon one side, and a corresponding diverging power upon the spectra on the other side. Let us suppose that the light is incident perpendicularly, and that the grating interval increases from the centre towards that edge which lies nearest to the spectrum under observation, and decreases towards the hinder edge.

It is evident that the waves from both halves of the grating are accelerated in an increasing degree, as we pass from the centre outwards, as compared with the phase they would possess were the central value of the grating interval maintained throughout. The irregularity of spacing has thus the effect of a convex lens, which accelerates the marginal relatively to the central rays. On the other side the effect is reversed. This kind of irregularity may clearly be present in a degree surpassing the usual limits, without loss of definition, when the telescope is focused so as to secure the best effect.

It may be worth while to examine further the other variations from correct ruling which correspond to the various terms expressing the deviation of the wave-surface from a perfect plane. If x and y be coordinates in the plane of the wave-surface, the axis of y being parallel to the lines of the grating, and the origin corresponding to the centre of the beam, we have as an approximate equation to the wave-surface (§ 6)

z== T p + ^ χ y i +^> +aχ3 +β χ2 y+yχf + bf+ . « (8); and, as we have just seen, the term in x 2 corresponds to a linear error in the spacing. In like manner, the term in y 2 corresponds to a general curvature of the lines (fig. 13), and does not influence the definition at the (primary) focus, although it may introduce astigmatism. If we suppose that everything is symmetrical on the two sides of the primary plane y = 0, the coefficients B, β, δ vanish. In spite of any inequality between p and ρ ', the definition will be good to this order of approximation, provided a and γvanish. The former measures the thickness of the primary focal line, and the latter measures its curvature. The error of ruling giving rise to a is one in which the intervals increase or decrease in both directions from the centre outwards (fig. 14), and it may often be compensated by a slight rotation in azimuth of the object-glass of the observing telescope. The term in y corresponds to a variation of curvature in crossing the grating (fig. 15).

When the plane zx is not a plane of symmetry, we have to consider the terms in xy, x 2 y, and y 3 . The first of these corresponds to a deviation from parallelism, causing the interval to alter gradually as we pass along the lines (fig. 16). The error thus arising may be compensated by a rotation of the object-glass about one of the diameters y= ± x . The term in x 2 y corresponds to a deviation from parallelism in the same direction on both sides of the central line (fig. 17); and that in y 3 would be caused by a curvature such that there is a point of inflexion at the middle of each line (fig. 18).

All the errors, except that depending on a, and especially those depending on γ and δ, can be diminished, without loss of resolving power, by contracting the vertical aperture. A linear error in the spacing, and a general curvature of the lines, are eliminated in the ordinary use of a grating.

The explanation of the difference of focus upon the two sides as due to unequal spacing was verified by Cornu upon gratings purposely constructed with an increasing interval. He has also shown how to rule a plane surface with lines so disposed that the grating shall of itself give well-focused spectra.

A similar idea appears to have guided Rowland to his brilliant invention of concave gratings, by which spectra can be photographed without any further optical appliance. In these instruments the lines are ruled upon a spherical surface of speculum metal, and mark the intersections of the surface by a system of parallel and equidistant planes, of which the middle member passes through the centre of the sphere. If we consider for the present only the primary plane of symmetry, the figure is reduced to two dimensions. Let AP (fig. 19) represent the surface of the grating, O being the centre of the circle. Then, if Q be any radiant point and Q' its image (primary focus) in the spherical mirror AP, we have

1 1 _ 2cos√> v 1 ^ r u a ’ where v 1 = AQ', u = AQ, α = OA, φ = angle of incidence QAO, equal to the angle of reflexion Q'AO. If Q be on the circle described upon OA as diameter, so that w = αcos<∕>, then Q' lies also upon the same circle; and in this case it follows from the symmetry that the unsymmetrical aberration (depending upon α) vanishes.

This disposition is adopted in Rowland’s instrument; only, in addition to the central image formed at the angle Φ'=Φ, there are a series of spectra with various values of φ', but all disposed upon the same circle. Rowland’s investigation is contained in the paper already referred to; but the following account of the theory is in the form adopted by Glazebrook.

In order to find the difference of optical distances between the courses QAQ', QPQ', we have to express QP-QA, PQ'-AQ'. To find the former, we have, if OAQ = ≠, AOP = ω,

QP 2 = tt 2 + 4α 2 sin 2 ⅜ω - 4αrtsin⅛ωsin(⅜ω - φ)

— (u + αsin≠sin ω) 2 - α 2 sin 2 ≠sin 2 ω + 4asin 2 ⅛ω (a - ttcos≠). Now as far as ω 4

4 sin 2 ⅜ft> = sin 2 ω +1 sin 4 ω, and thus to the same order

QP 2 = (u + αsin <∕>sin ω) 2 - a cos φ (u - a cos φ) sin 2 ω + ∣α (oe - u cos φ) sin i ω .

But if we now suppose that Q lies on the circle w = αcosφ, the middle term vanishes, and we get, correct as far as ω 4 , ∕λt> ∕ . · · ∖ ∕ ( , α 2 sin 2 <⅛sin 4 ω )

QP = (tt + αsιn≠smω) ∕ 1 1-∣ f·

V ( 4u )

so that

QP-∙u=αsin≠sino> + ∣αsin<∕>tan≠sin 4 ω . . . (9), in which it is to be noticed that the adjustment necessary to secure the disappearance of sin 2 w is sufficient also to destroy the term in sin 3 ω.

A similar expression can be found for Q'P - Q'A; and thus, if Q'A = v, Q'ΛO = ≠', where ν = a cos φ', we get

QP + PQ' - Q A - AQ' = a sin a> (sin φ - sin φ , )

+ ∣αsin 4 ω(sin≠tan<∕> + sinφ'tan<∕>') . . . (10).

If Φ' ≈φ, the term of the first order vanishes, and the reduction of the difference of path via P and via A to a term of the fourth order proves not only that Q and Q' are conjugate foci, but also that the foci are exempt from the most important term in the aberration. In the present application φ' is not necessarily equal to φ ; but if P correspond to a line upon the grating, the difference of retardations for consecutive positions of P, so far as expressed by the term of the first order, will be equal to ψwtλ (in integral), and therefore without influence, provided

σ(sin≠-sin≠')=ψwιλ (ll), where σ denotes the constant interval between the planes containing the lines. This is the ordinary formula for a reflecting plane grating, and it shows that the spectra are formed in the usual directions. They are here focused (so far as the rays in the primary plane are concerned) upon the circle OQ'A, and the outstanding aberration is of the fourth order.

In order that a large part of the field of view may be in focus at once, it is desirable that the locus of the focused spectrum should be nearly perpendicular to the line of vision. For this purpose Rowland places the eye-piece at 0, so that ≠ = 0, and then by (11) the value of Φ' in the m th spectrum is

<rsin≠'=±mλ (12).

If ω now relate to the edge of the grating, on which there are altogether n lines, nσ=2αsinw, and the value of the last term in (10) becomes

- r ⅛π<r sin 3 ω sin φ' tan φ', or τ ⅛∙7nnλsin 3 ωtan≠' (13).

This expresses the retardation of the extreme relatively to the central ray, and is to be reckoned positive, whatever may be the signs of ω and φ ’ . If the semi-angular aperture (co) be τ ⅜< 5 -, and tan φ'=l, mn might be as great as four millions before the error of phase would reach ⅜λ. If it were, desired to use an angular aperture so large that the aberration according to (13) would be injurious, Rowland points out that on his machine there would be no difficulty in applying a remedy by making σ slightly variable towards the edges. Or, retaining σ constant, we might attain compensation by so polishing the surface as to bring the circumference slightly forward in comparison with the position it would occupy upon a true sphere.

It may be remarked that these calculations apply to the rays in the primary plane only. The image is greatly affected with astigmatism; but this is of little consequence, if γ in (8) be small enough. Curvature of the primary focal line having a very injurious effect upon definition, it may be inferred from the excellent performance of these gratings that 7 is in fact small. Its value does not appear to have been calculated. The other coefficients in (8) vanish in virtue of the symmetry.

The mechanical arrangements for maintaining the focus are of great simplicity. The grating at A and the eye-piece at O are rigidly attached to a bar AO, whose ends rest on carriages, moving on rails OQ, AQ at right angles to each other. A tie between C and Q can be used if thought desirable.

The absence of chromatic aberration gives a great advantage in the comparison of overlapping spectra, which Rowland has turned to excellent account in his determinations of the relative wavelengths of lines in the solar spectrum.

For absolute determinations of wave-lengths plane gratings are used. It is found that the angular measurements present less difficulty than the comparison of the grating interval with the standard metre. There is also some uncertainty as to the actual temperature of the grating when in use. In order to minimize the heating action of the light, it might be submitted to a preliminary prismatic analysis before it reaches the slit of the spectrometer, after the manner of Von Helmholtz (Optics, vol. xvii. p. 802).

Bell found further that it is necessary to submit the gratings to calibration, and not to rest satisfied with a knowledge of the number of lines and of the total width. It not unfrequently happens that near the beginning of the ruling the interval is anomalous. If the width of this region be small, it has scarcely any effect upon the angular measurements, and should be left out of account in estimating the effective interval.

§15. Theory of Corrugated Waves.

The theory of gratings is usually given in a form applicable only to the case where the alternate parts are transparent and opaque. Even then it is very improbable that the process of simply including the transparent parts and excluding the opaque parts in the integrations of § 11 gives an accurate result. The condition of things in actual gratings is much more complicated, and all that can with confidence be asserted is the approximate periodicity in the interval σ. The problem thus presents itself—to determine the course of events on the further side of the plane z = 0 when the amplitude and phase over that plane are periodic functions of x; and the first step in the solution would naturally be to determine the effect corresponding to the infinitesimal strip ydx over which the amplitude and phase are constant. In fig. 20 QQ' represents the strip in question, of which the effect is to be estimated at P(0, 0, z ); QR = y, RP= r , QP = ρ.

If we assume the law of secondary wave determined in § 10 so as to suit the resolution of an infinite uniform primary wave, we have, as the effect of QQ', 2dx f cos κ(αi-p + 1λ)

Jo hP 2dx f™ dp . , . ...

^^^r√∙ √⅛⅛> sιn ' ,t ' ,i ^ ri (1> ·

The development of this expression for the operation of a linear source would take us too far. We must content ourselves with the limiting form assumed when κr is great, as it would almost always be in optics. Under these circumstances the denominator may be simplified by writing

√(p 2 - r 2 ) = √(2r). √(p - r), so that (1) becomes 2dx f cο d(p-r) ,

√(p-U7) sιn √^- i -(p-^)}∙

Now

∕* e0 sinκ24du _ ∕* x cos κudu _ ∕∕ π ∖ _

Jo ^fu Jo f Ju ~~ ∖ ∖2κ∕ ~ 2√ λ , and thus we obtain - √(⅛7) { sin κ ^ at ~ r ' , ~ cos κ ^ at ^ ifø . , , ,

= -√(^) sιnκ ^" r -^ · · · ∙.∙ (2 )> which gives the effect of a linear source at a great distance. The occurrence of the factor r -½ is a consequence of the cylindrical expansion of the waves. The whole effect is retarded one-eighth of a period in comparison with that of the central element, instead of one-quarter of a period as in the case of a uniform wave extending over the whole plane.

The effect of the uniform plane wave can be recovered by integrating (2) with respect to x from - ∞ to + ∞, on the supposition that κr is great. We have dx _ rdr _ 'Jr.d(r -z) _

√r - Jr.x √(r + z).√(r-z) ’

and in this, since the only elements which contribute sensibly to the integral are those for which (r - z) is small, we may write √r 1

√(r + z) √2 ’

The integral can then be evaluated by the same formula as before, aud we get finally cosκ (at-z), the same as if the primary wave were supposed to advance without resolution. The recomposition of the primary wave by integration with rectangular coordinates is thus verified, but only under the limitation, not really required by the nature of the case, that the point at which the effect is to be estimated is distant by a very great number of wave-lengths from the plane of resolution.

We will now suppose that the amplitude and phase of the primary wave at the plane of resolution z = 0 are no longer constants, but periodic functions of x. Instead of cos καt simply, we should have to take in general

A cos (px +f) cos κat + B cos (px + g) sin κ at; but it will be sufficient for our purpose to consider the first term only, in which we may further put for simplicity A=1, f =0. The effect of the linear element at x, 0, upon a point at ξ, z, will be, according to (2), i^-rC°snz sin κ(αi - r - 1λ),

√(λr)

where r is the distance, expressed by r 2 =z 2 + (x- ξ) 2 · Thus, if we write x=ξ + a, the whole effect is

∖(V, {s⅛(∞'+M-

+ sin (καί-χ-⅛τr-κr-ya)} . . (3), where r 2 = z 2 + α 2 .

In the two terms of this integral the elements are in general of rapidly fluctuating sign; and the only important part of the range of integration in (for example) the first term is in the neighbourhood of the place where pa - κr is stationary in value, or where pda-κdr=0 (4).

In general ada-rdr=0, so that if the values of a and r corresponding to (4) be called α 0 , r 0 , we have α 0 =s Λ> = ≡ z-> p κ √(κ 2 -2> 2 ) ' ’·

Now, in the neighbourhood of these values, if α = α 0 + α 1 , ya - κr-ρa, - κr 0 +ajρ - √S) - »’(ί -£) , in which by (5) the term of the first order vanishes. Using this in (3), we get for the first term c °2√(λD 1 sin ^ κat+2 ^ ~⅛ π ~ κr ° +ρa °) cosλaι2

- cos (καί +pξ - ∣7Γ - κr 0 +pa 0 )sinΛa 1 2 J, where for brevity h is written for

2⅞(1 "⅛)'

The integration is effected by means of the formula cοshu 2 du =smhu 2 du = (Tf); and we find 2 √( κ 2--pj C0S {Kat - Kr 0 + ·

The other term in (3) gives in like manner

2∖∕(κ 2 -p 2 ) C0S ~ κr ° ^*^7 la o) ί

so that the complete value is Kai "^ ( ' c2 "·P 2) · z }· · · (6 >∙

When p = 0, we fall back on the uniform plane wave travelling with velocity a . In general the velocity is not a , but

κa∕√(κ 2 -jn 2 ) (7).

The wave represented by (6) is one in which the amplitude at various points of a wave-front is proportional to cosρξ, or cosjxc; and, beyond the reversals of phase herein implied, the phase is constant, so that the wave-surfaces are given by z= constant. The wave thus described moves forward at the velocity given by (7), and with type unchanged.

The above investigation may be regarded as applicable to gratings which give spectra of the first order only. Although κ vary, there is no separation of colours. Such a separation requires either a limitation in the width of the grating (here supposed to be infinite), or the use of a focusing lens.

It is important to remark that p has been assumed to be less than κ, or σ greater than λ; otherwise no part of the range of integration in (3) is exempt from rapid fluctuation of sign, and the result must be considered to be zero. The principle that irregularities in a wave-front of periods less than λ cannot be propagated is of great consequence. Further light will be thrown upon it by a different investigation to be given presently.

The possibility of the wave represented by (6) is perhaps sufficiently established by the preceding method, but the occurrence of the factor κ∕∖∕(κ 2 -p 2 ) shows that the laws of the secondary wave (determined originally from a consideration of uniform plane waves) was not rightly assumed.

The correct law applicable in any case may be investigated as follows. Let us assume that the expression for the wave of given periodic time is

≠ = Λ^- 1 e - i ' c PF(a ≈,y)d≈dy 5 . . . (8), and let us inquire what the value of F (x, y) must be in order that the application of Huygens’s principle may give a correct result. From(8)

and £ ( e ~ iκρ ∖ e~ fe φ + κp) - dp∖ ρ J p 2

We propose now to find the limiting value of dψ∣dz when z is very small. The value of the integral will depend upon those elements only for which x and y are very small, so that we replace F(x, y) in the limit by F(0, 0). Also, in the limit, ff z f τf~r) ,lxιl ' j -ff so that

Limit t ⅛=-2τr∕ καi F(0, 0).

dz

The proper value of e i * αi F(z, ι∕) is therefore that of -dty∣dz at the same point (x, y, 0) divided by 2π, and we have in general

→∕(f) s 7≡** · · · · »

In the case of the uniform plane wave,

⅛(αi-2) . iκ(at-z) ψ=--c , aψ∕az=-2κe; so that ic iκat .i κ ( at -p+∖λ) *-× JJ ~r ixiv JJ—τ t — dxi,j ' agreeing with what we have already found for the secondary wave in this case.

But, if ψ= cos px. e i t κal V(κ 2 p 2 )∙z], f ⅛ (z=0) = - t√(κ 2 -p 2 ) cospxe κal ', az and ♦ - V( ⅛¾ z ^ ,ιw P~ c ° ay ^^ ι ' · The occurrence of the anomalous factor in (6) is thus explained.

It must be admitted that the present process of investigation is rather artificial; and the cause lies in the attempt to dispense with the differential equation satisfied by ψ, viz.,

S + O + S ,+ * 2ψ ~ o (10) ’

on which in the case of sound the whole theory is based. It is in fact easy to verify that any value of ψ included under (8), where ρ 3 =(ξ-x) 2 +(n-y) 2 + Ϛ 2 , satisfies the equation

S' + O + S + * 2ψ = o ∙

When there is no question of resolution by Huygens’s principle, the distinction between ξ, y and x, y may be dropped.

Starting from the differential equation, we may recover previous results very simply. If ψ be proportional to cos px ; cos qy , we have

⅛+(κ 2 √√)Ψ=θ (11).

If κ 2 -p 2 -⅛ 2 -=∕t 2 , μ being real, the solution of (11) is

ψ = AΛ z + Be-^, where A and B are independent of z. Restoring the factors involving t, X, y, we may write ψ = cos ρx cos qy( a f κat + μz ) + B c i <∙ κat ~ μ2 >} · (12), of which the first term may be dropped when we contemplate waves travelling in the positive direction only. The corresponding realized solution is of the type ψ =cos px cos qy cos {καί-∖∕(κ 2 -p 2 - 9 2 ).2} . . (13).

When κ 2 > (p 2 + q 2 ), the wave travels without change of type and with velocity xr κa ∖∕(κ 2 -p 2 -q 2 ) (14)∙

We have now to consider what occurs when κ 2 <(p 2 + ^ 2 ). If we write κ 2 -p 2 -q 2 = - μ 2 , we have in place of (12)

ψ = cospxcosg∙y[Ae iκαi ^ ’ ^ μz +Be* κai-μ2 }. · · (15); and for the realized solution corresponding to (13)

ψ = cospx cos qy e~ fiz cos κat (16).

We conclude that under these circumstances the motion rapidly diminishes as z increases, and that no wave in the usual sense can be propagated at all.

It follows that corrugations of a reflecting surface (no matter how deep) will not disturb the regularity of a perpendicularly reflected wave, provided the wave-length of the corrugation do not exceed that of the vibration. And, whatever the former wave-length may be in relation to the latter, regular reflexion will occur when the incidence is sufficiently oblique.

The first form of solution may be applied to give an explanation of the appearances observed when a plane wave traverses a parallel coarse grating and then impinges upon a screen held at varying distances behind. As the general expression of the wave periodic with respect to x in distance σ we may take

A o cos (κat - κz) + A 1 cos (px +∕ 1 ) cos (καί - μ 1 z)

+ B 1 cos(pz + y 1 ) sin (καί-μ 1 z) 4-A 2 cos(¾w+∕ 2 ) cos κ(αi - μ s z)+ ..., where p = 2π∕<r, κ = 2τr∣K, and μ 1 2 =κ 2 -p 2 , μ s 2 = κ 2 -4p 2 , . . ., the series being continued as long as μ is real. We shall here, however, limit ourselves to the first three terms, and in them suppose A 1 and B 1 to be small relatively to A 0 . The intensity may then be represented by

A 0 2 + 2 A o A 1 cos (px +∕) cos (κz - j u 1 z)

4-2A 0 B 1 cos(2∞ + <7)sin(κz- j u 1 z) .... (17).

The stripes thrown upon the screen in various positions are thus periodic functions of z, and the period is 2τr 2σ 2

K-√(κ 2 - - p 2 )~ A, k ,, if λ be supposed small in comparison with σ. It may be noticed that, if the position of the screen be altered by the half of this amount, the effect is equivalent to a shifting parallel to x through the distance ½σ. Hence, if the grating consists of alternate transparent and opaque parts of width ½σ, the stripes seen upon the screen are reversed when the latter is drawn back through the distance σ 2 ∕λ. In this case we may suppose B 1 to vanish, and (17)

then shows that the field is uniform when the screen occupies positions midway between those which give the most distinct patterns. These results are of interest in connexion with the photographic reproduction of gratings.

§ 16. Talbot's Bands.

These very remarkable bands are seen under certain conditions when a tolerably pure spectrum is regarded with the naked eye, or with a telescope, half the aperture being covered by a thin plate, e.g., of glass or mica. The view of the matter taken by the discoverer was that any ray which suffered in traversing the plate a retardation of an odd number of half wave-lengths would be extinguished, and that thus the spectrum would be seen interrupted by a number of dark bars. But this explanation cannot be accepted as it stands, being open to the same objection as Arago’s theory of stellar scintillation. It is as far as possible from being true that a body emitting homogeneous light would disappear on merely covering half the aperture of vision with a half-wave plate. Such a conclusion would be in the face of the principle of energy, which teaches plainly that the retardation in question leaves the aggregate brightness unaltered. The actual formation of the bands comes about in a very curious way, as is shown by a circumstance first observed by Brewster. When the retarding plate is held on the side towards the red of the spectrum, the bands are not seen. Even in the contrary case, the thickness of the plate must not exceed a certain limit, however pure the spectrum may be. A satisfactory explanation of these bands was first given by Airy, but we shall here follow the investigation of Stokes, limiting ourselves, however, to the case where the retarded and unretarded beams are contiguous and of equal width. The aperture of the unretarded beam may thus be taken to be limited by - h, x =0, y = -l, y= +l ; and that of the beam retarded by R to be given by x =0, x=h, y= - I, y= +l. For the former (1) § 11 gives

1 Z* 0 ∕*÷ i · f , ∕∙⅛÷!∕Ι( j j

5 ∕ ∕ sιnκj at-f+ * ,y } dxdy ^JJ-hJ-l I J J 2lh f , κyl 2f κξh ί . j. ch Ί

--v∙⅛ sm τ∙⅛ s,n i∙ s,n d ai -∙ z 2∕} · < 1 >, on integration and reduction.

For the retarded stream the only difference is that we must subtract R from at, and that the limits of x are 0 and + h . We thus get for the disturbance at ξ, η due to this stream 2lh f . κyl 2f . κch . f . ch j " v ’ ^ sιn 7"^ sιn 17' smκ ι αi ^ z-β+ 2r∕ · (2) · If we put for shortness τ for the quantity under the last circular function in (1), the expressions (1), (2) may be put under the forms u sinτ, v sin(τ - α) respectively; and, if I be the intensity, I will be measured by the sum of the squares of the coefficients of sin τ and cos τ in the expression u sin τ + v sin (τ - α), so that

I = u 2 + v 2 + 2uν cos a, which becomes on putting for u, v, and a their values, and putting {.⅛≡T} ’ → < 3 >∙

l-Q∙⅛s⅛ 2 ^{2 + 2cos(-√-v)} · · <⅛ If the subject of examination be a luminous line parallel to y, we shall obtain what we require by integrating (4) with respect to ηfrom - ∞ to + ∞ . The constant multiplier is of no especial interest, so that we may take as applicable to the image of a line τ 2 . „ πξh ί , ∕ 2τrR 2πξh ∖ ) I_ ? s “Vi 1+cos (^v-Ι√)} · < 5 >∙

If R=½λ, I vanishes at ξ = 0; but the whole illumination, repre- ∕*+∞

sented by/ I dξ, is independent of the value of R. If R = 0, 1= ~ 2 sin 2 -^J-> in agreement with § 11, where a has the meaning here attached to 2h.

The expression (5) gives the illumination at ξ due to that part of the complete image whose geometrical focus is at ξ = 0, the retardation for this component being R. Since we have now to integrate for the whole illumination at a particular point O due to all the components which have their foci in its neighbourhood, we may conveniently regard O as origin. ξ is then the coordinate relatively to O of any focal point O' for which the retardation is R; and the required result is obtained by simply integrating (5) with respect to ξ from - ∞ to + ∞. To each value of ξ corresponds a different value of λ, and (in consequence of the dispersing power of the plate) of R. The variation of λ may, however, be neglected in the integration, except in 2πR∕λ, where a small variation of λ entails a comparatively large alteration of phase. If we write

P = 2πR∕λ (6), we must regard p as a function of ξ, and we may take with sufficient approximation under any ordinary circumstances p-p' + wξ (7), where ρ ' denotes the value of ρ at O, and ῶ is a constant, which is positive when the retarding plate is held at the side on which the blue of the spectrum is seen. The possibility of dark bands depends upon ῶ being positive. Only in this case can cos(p ’ ÷ (w — 2πlιjλf)ξ } retain the constant value - 1 throughout the integration, and then only when w=2πhι∕λf (8), and cos p' = -1 (9).

The first of these equations is the condition for the formation of dark bands, and the second marks their situation, which is the same as that determined by the imperfect theory.

The integration can be effected without much difficulty. For the first term in (5) the evaluation is effected at once by a known formula. In the second term if we observe that cos{p' + (ze - 2πh∣κf) ξ} = cos{ p ’- g i ξ} = cos p'cos g 1 ξ + sin p' sin g 1 ξ , we see that the second part vanishes when integrated, and that the remaining integral is of the form w=∕* + °°sin 2 7⅛cosy 1 ξ ∣∣, √-∞ ξ

where

λ 1 = π7t∕λ∕, g 1 =zr -2πh∣Kf .... (10).

By differentiation with respect to g x it may be proved that w = 0 from g 1 = - ∞ to g 1 =-2h 1 , w = ⅛ιr(2h 1 + g 1 ) from <∕ 1 =-2A 1 to <7 1 = 0, w=⅛π(2h 1 -gj from gι= 0 to g 1 = 2Λ 1 , w=0 from <∕ 1 = 27⅛ 1 to <∕ j = ∞ .

The integrated intensity, 1', or

2π7t 1 + 2 cos p w , is thus

Γ = 2ιrΛ 1 (ll), when g 1 numerically exceeds 27t 1 ; and, when g 1 lies between ± 2 h 1 ,

I = π {∙2Λ 1 + (27t 1 - √<∕ 1 2 ) cos p'} .... (12).

It appears therefore that there are no bands at all unless zr lies between 0 and + 4 h 1 , and that within these limits the best bands are formed at the middle of the range when ῶ =2h 1 . The formation of bands thus requires that the retarding plate be held upon the side already specified, so that ῶ be positive; and that the thickness of the plate (to which ῶ is proportional) do not exceed a certain limit, which we may call 2T 0 . At the best thickness T 0 the bands are black, and not otherwise.

The linear width of the band (e )is the increment of ξ which alters p by 2π, so that e = 2π ∣ῶ (13).

With the best thickness ῶ=2πh∣λf (14), so that in this case e=λ∕∕7⅛, (15).

The bands are thus of the same width as those due to two infinitely narrow apertures coincident with the central lines of the retarded and unretarded streams, the subject of examination being itself a fine luminous line.

If it be desired to see a given number of bands in the whole or in any part of the spectrum, the thickness of the retarding plate is thereby determined, independently of all other considerations. But in order that the bands may be really visible, and still more in order that they may bc black, another condition must be satisfied. It is necessary that the aperture of the pupil be accommodated to the angular extent of the spectrum, or reciprocally. Black bands will be too fine to be well seen unless the aperture (2 h ) of the pupil be somewhat contracted. One-twentieth to one-fiftieth of an inch is suitable. The aperture and the number of bands being both fixed, the condition of blackness determines the angular magnitude of a band and of the spectrum. The use of a grating is very convenient, for not only are there several spectra in view at the same time, but the dispersion can be varied continuously by sloping the grating. The slits may be cut out of tin-plate, and half covered by mica or “microscopic glass,” held in position by a little cement.

If a telescope be employed there is a distinction to be observed, according as the half-covered aperture is between the eye and the ocular, or in front of the object-glass. In the former case the function of the telescope is simply to increase the dispersion, and the formation of the bands is of course independent of the particular manner in which the dispersion arises. If, however, the half-covered aperture be in front of the object-glass, the phenomenon is magnified as a whole, and the desirable relation between the (unmagnified) dispersion and the aperture is the same as without the telescope. There appears to be no further advantage in the use of a telescope than the increased facility of accommodation, and for this of course a very low power suffices.

The original investigation of Stokes, here briefly sketched, extends also to the case where the streams are of unequal width h, k, and are separated by an interval 2g. In the case of unequal width the bands cannot be black; but if h=k, the finiteness of 2g does not preclude the formation of black bands.

The theory of Talbot’s bands with a half-covered circular aperture has been treated by H. Struve.

§ 17. Diffraction when the Source of Light is not Seen in Focus.

The phenomena to be considered under this head are of less importance than those investigated by Fraunhofer, and will be treated in less detail; but, in view of their historical interest and of the ease with which many of the experiments may be tried, some account of their theory could not be excluded from such a work as the present. One or two examples have already attracted our attention when considering Huygens’s zones, viz., the shadow of a circular disk, and of a screen circularly perforated; but the most famous problem of this class—first solved by Fresnel—relates to the shadow of a screen bounded by a straight edge.

In theoretical investigations these problems are usually treated as of two dimensions only, everything being referred to the plane passing through the luminous point and perpendicular to the diffracting edges, supposed to be straight and parallel. In strictness this idea is appropriate only when the source is a luminous line, emitting cylindrical waves, such as might be obtained from a luminous point with the aid of a cylindrical lens. When, in order to apply Huygens’s principle, the wave is supposed to be broken up, the phase is the same at every element of the surface of resolution which lies upon a line perpendicular to the plane of reference, and thus the effect of the whole line, or rather infinitesimal strip, is related in a constant manner (§ 15) to that of the element which lies in the plane of reference, and may be considered to be represented thereby. The same method of representation is applicable to spherical waves, issuing from a point, if the radius of curvature be large; for, although there is variation of phase along the length of the infinitesimal strip, the whole effect depends practically upon that of the central parts where the phase is sensibly constant.

In fig. 21 APQ is the arc of the circle representative of the wave-front of resolution, the centre being at O, and the radius OA being equal to a. B is the point at which the effect is required, distant a+b from O, so that AB = b ; AP = s , PQ = ds .

Taking as the standard phase that of the secondary wave from A, we may represent the effect of PQ by cos 2π ∖ - - - ). ds , ∖τ hJ where δ — BP - AP is the retardation at B of the wave from P relatively to that from A.

Now

δ ~ (α + δ)s 2 ∕2αδ ( 1 ) j so that, if we write

2πδ π(α + δ)s 2 τr „

^χ - 2*^ (2).

the effect at B is

'( 2(S⅛ 1 1 { cββ ΊΓ.- 7rr5 ∙ rf "' +sin V f sin 4lrt * dv } W* the limits of integration depending upon the disposition of the diffracting edges. When a, b, λ are regarded as constant, the first factor may be omitted,—as indeed should be done for consistency’s sake, inasmuch as other factors of tho same nature have been omitted already.

The intensity I 2 , the quantity with which we are principally concerned, may thus be expressed

I 2 = {∕'cos¾7rr> 2 .dv} 2 +{y^sin¾7τv 2 .dv} 2 . . . (4).

These integrals, taken from v =0, are known as Fresnel’s integrals; we will denote them by C and S, so that

C==∕* ι cos ⅜πr 2 .i⅛, S= ∕* t sin ⅛πv 2 .dυ . . . (5).

√0 √0

When the upper limit is infinity, so that the limits correspond to the inclusion of half the primary wave, C and S are both equal to ½, by a known formula; and on account of the rapid fluctuation of sign the parts of the range beyond very moderate values of υ contribute but little to the result.

Ascending series for C and S were given by Knockenhauer, and are readily investigated. Integrating by parts, we find r , ∙o f v i ∙i πι2 , 1 *∙1ιr∣>≡ . r v i.^πυ 2 o

C + iS=∕ e dv≈e .v~liπ∕ e dv 3 ∙, fo fo and, by continuing this process, c + ^∙ i "T-ι' s ÷⅜⅞^¾⅞U 7 ÷∙∙∙i∙

By separation of real and imaginary parts,

C = M cos ⅜πv 2 + N sin ⅜πt- 2 ^∣ , fi .

S = M sin ⅜π√ 2 - N cos ⅛ιri> 2 J ' , where , r v π⅞ 5 ι π 4 υ 9 M== Ï"3^ + 3XÜ9" < 7) ’

„ πv 3 π 3 v 7 ιr s V n

~L3~1.3.5.7 + 1.3.5.7.9.11 · · · · (s )∙

These series are convergent for all values of v, but are practically useful only when v is small.

Expressions suitable for discussion when v is large were obtained by Gilbert. Taking

⅜πt> 2 = w (9), we may write ∕∙u e iu. 1

C + fS=√-∕ (10).

Again, by a known formula,

JL„JL Γl¾ (u) √tt ∖∕τrf( i -√iC

Substituting this in (10), and inverting the order of integration, we get c+is=-⅛∕'° 0 ⅛ ∕V ci -‰

7γ √ 2 √o ≠ fo 1 ∕∙° 0 dx e u < t '- χ )-l π V 2 Jo ^√' χ ' i-χ

Thus, if we take

1 f x e- vx ∖∕x.dx „ 1 ∕* c0 e~ υχ dx

π√2√ 0 1+ae 2 ’ π√2√ 0 √a√l+z 2 ) * i λ

C = ⅜-G cosu + H sinu, S = ⅜ - G sin u - II cosu . (14).

The constant parts in (14), viz. ½, may be determined by direct integration of (12), or from the observation that by their constitution G and H vanish when u=∞, coupled with the fact that C and S then assume the value ½.

Comparing the expressions for C, S in terms of M, N, and in terms of G, H, we find that

G=⅜½(cos w-f-sin u) - M, H = ⅛(cos « - sin u) + N . (15), formulae which may be utilized for the calculation of G, II when u (or 1?) is small. For example, when u = 0, M = 0, N = 0, and consequently G = H = ½.

Descending series of the semi-convergent class, available for numerical calculation when u is moderately large, can be obtained from (12) by writing x=uy, and expanding the denominator in powers of y. The integration of the several terms may then be effected by the formula J e V~ σ dy=Γ(2' + i) = ⅛-⅛)⅛-⅜). .. ⅛√πj fo and we get in terms of v n 1 1.3.5 , 1.3.5.7.9

π⅞ 3 π⅛∙ 7 + ιr⅛ u · · · · (lθ), lj 1 1.3 1.3.5.7

τrv jγM π 5 ⅛' 9 (^)∙

The corresponding values of C and S were originally derived by Cauchy, without the use of Gilbert’s integrals, by direct integration by parts.

From the series for G and H just obtained it is easy to verify that tZH rι dG ττ , ,, nx

-^=-πrG, ^=πt∙H-l .... (18).

We now proceed to consider more particularly the distribution of light upon a screen PBQ near the shadow of a straight edge A. At a point P within the geometrical shadow of the obstacle, the half of the wave to the right of C (fig. 22), the nearest point on the wave-front, is wholly intercepted, and on the left the integration is to be taken from s = CA to s = ∞. If V be the value of υ corresponding to CA, viz., v -√P-⅛⅛ a ∙ < ιs >- we may write

-oe cos ⅜πr 2 .cZυ MX” sin ⅜7rυ 2 .dv^ . . (20), or, according to our previous notation,

I 2 -(⅛-G v ) 2 +(⅜-S v ) 2 ~G 2 + H 2 . . . . (21).

Now in the integrals represented by G and H every element diminishes as V increases from zero. Hence, as CA increases, viz., as the point P is more and more deeply immersed in the shadow, the illumination continuously decreases, and that without limit. It has long been known from observation that there are no bands on the interior side of the shadow of the edge.

The law of diminution when V is moderately large is easily expressed with the aid of the series (16), (17) for G, H. We have ultimately G = 0, H=(πV) -1 , so that

I 2 = l∕π 2 V 2 , or the illumination is inversely as the square of the distance from the shadow of the edge.

For a point Q outside the shadow the integration extends over more than half the primary wave. The intensity may be expressed by I 2 = (⅜ + C v ) 2 + (⅛ + Sv) 2 (22); and the maxima and minima occur when . 1 _ , dC ., γi , dS n

(⅜ + Cv) + (⅜ + Sv) ^γ = θ, whence sin ⅜πV 2 +cos ⅛ιrV 2 = G (23).

When V=0, viz., at the edge of the shadow, I a = ⅜; when V = ∞, I 2 = 2, on the scale adopted. The latter is the intensity due to the uninterrupted wave. The quadrupling of the intensity in passing outwards from the edge of the shadow is, however, accompanied by fluctuations giving rise to bright and dark bands. The position of these bands determined by (23) may be very simply expressed when V is large, for then sensibly G = 0, and

⅛πV 2 = fτr + nτr (24), n being an integer. In terms of δ, we have from (2)

δ = (f + ⅞n)λ (25).

The first maximum in fact occurs when δ = ⅜λ — ⋅0046λ, and the first minimum when δ = ⅞λ-0016λ, the corrections being readily obtainable from a table of G by substitution of the approximate value of V.

The position of Q corresponding to a given value of V, that is, to a band of given order, is by (19)

. . (26).

By means of this expression we may trace the locus of a band of given order as b varies. With sufficient approximation we may regard BQ and b as rectangular coordinates of Q. Denoting them by X, y, so that AB is axis of y and a perpendicular through A the axis of X, and rationalizing (26), we have

2αx 2 - V 2 λy 2 - V 2 αλy = 0, which represents a hyperbola with vertices at 0 and A.

From (24), (26) we see that the width of the bands is of the order √{ δλ(α + 5)∕α}. From this we may infer the limitation upon the width of the source of light, in order that the bands may be properly formed. If ω be the apparent magnitude of the source seen from A, ωb should be much smaller than the above quantity, or .u< √{λ(α + δ)∕αδ} (27),

If a be very great in relation to b, the condition becomes ω<√(λ∕δ) (28), so that if b is to be moderately great (1 metre), the apparent magnitude of the sun must be greatly reduced before it can be used as a source.

The values of V for the maxima and minima of intensity, and the magnitudes of the latter, were calculated by Fresnel. An extract from his results is given in the accompanying table. [table]

A very thorough investigation of this and other related questions, accompanied by fully worked-out tables of the functions concerned, will be found in a recent paper by Lommel.

When the functions C and S have once been calculated, the discussion of various diffraction problems is much facilitated by the idea, due to Cornu, of exhibiting as a curve the relationship between C and S, considered as the rectangular coordinates (x, y) of a point. Such a curve is shown in fig. 23, where, according to the definition (5) of C, S, at=y∕* cos⅛π√ 1 .ifo, sιn ⅛ πy2 ∙ c l v · · (29).

The origin of coordinates 0 corresponds to v= 0; and the asymptotic points J, J', round which the curve revolves in an ever-closing spiral, correspond to υ= ±∞.

The intrinsic equation, expressing the relation between the arc σ (measured from 0) and the inclination φ of the tangent at any point to the axis of x, assumes a very simple form. For dx ≈ cos ⅜πu 2 . dv, dy = sin ⅜π v 2 . dv; so that

<r=y √(dx 2 + dy) = v (30),

Φ= tan-1 ⅛=i^ (31).

Accordingly,

≠ = ⅜ιro- 2 (32); and for the curvature,

S = - ... (33).

Cornu remarks that this equation suffices to determine the general character of the curve. For the osculating circle at any point includes the whole of the curve which lies beyond; and the successive convolutions envelop one another without intersection.

The utility of the curve depends upon the fact that the elements of arc represent, in amplitude and phase, the component vibrations due to the corresponding portions of the primary wave-front. For by (30) dσ = dv , and by (2) dv is proportional to ds. Moreover by (2) and (31) the retardation of phase of the elementary vibration from PQ (fig. 21) is 2πδ∕λ, or φ. Hence, in accordance with the rule for compounding vector quantities, the resultant vibration at B, due to any finite part of the primary wave, is represented in amplitude and phase by the chord joining the extremities of the corresponding arc (σ 2 - σ 1 ).

In applying the curve in special cases of diffraction to exhibit the effect at any point P (fig. 22) the centre of the curve O is to be considered to correspond to that point C of the primary wave-front which lies nearest to P. The operative part, or parts, of the curve are of course those which represent the unobstructed portions of the primary wave.

Let us reconsider, following Cornu, the diffraction of a screen unlimited on one side, and on the other terminated by a straight edge. On the illuminated side, at a distance from the shadow, the vibration is represented by JJ'. The coordinates of J, J' being (½, ½), ( - ½, - ½), I 2 is 2; and the phase is ⅛ period in arrear of that of the element at O. As the point under contemplation is supposed to approach the shadow, the vibration is represented by the chord drawn from J to a point on the other half of the curve, which travels inwards from J' towards O. The amplitude is thus subject to fluctuations, which increase as the shadow is approached. At the point O the intensity is one-quarter of that of the entire wave, and after this point is passed, that is, when we have entered the geometrical shadow, the intensity falls off gradually to zero, without fluctuations. The whole progress of the phenomenon is thus exhibited to the eye in a very instructive manner.

We will next suppose that the light is transmitted by a slit, and inquire what is the effect of varying the width of the slit upon the illumination at the projection of its centre. Under these circumstances the arc to be considered is bisected at O, and its length is proportional to the width of the slit. It is easy to see that the length of the chord (which passes in all cases through O) increases to a maximum near the place where the phase-retardation is ⅜ of a period, then diminishes to a minimum when the retardation is about ⅞ of a period, and so on.

If the slit is of constant width and we require the illumination at various points on the screen behind it, we must regard the arc of the curve as of constant length. The intensity is then, as always, represented by the square of the length of the chord. If the slit be narrow, so that the arc is short, the intensity is constant over a wide range, and does not fall off to an important extent until the discrepancy of the extreme phases reaches about a quarter of a period.

We have hitherto supposed that the shadow of a diffracting obstacle is received upon a diffusing screen, or, which comes to nearly the same thing, is observed with an eye-piece. If the eye, provided if necessary with a perforated plate in order to reduce the aperture, be situated inside the shadow at a place where the illumination is still sensible, and be focused upon the diffracting edge, the light which it receives will appear to come from the neighbourhood of the edge, and will present the effect of a silver lining. This is doubtless the explanation of a “pretty optical phenomenon, seen in Switzerland, when the sun rises from behind distant trees standing on the summit of a mountain.”

§ 18. Diffraction Symmetrical about an Axis.

The general problem of the diffraction pattern due to a source of light concentrated in a point, when the system is symmetrical about an axis, has been ably investigated by Lommel. We must content ourselves here with a very slight sketch of some of his results.

Spherical waves, centred upon the axis, of radius a fall upon the diffracting screen; and the illumination is required on a second screen, like the first perpendicular to the axis, at a distance (a + b ) from the source. We have first to express the distance (d ) between an element dS of the wave-front and a point M in the plane of the second screen. Let ζ denote the distance of M from the axis of symmetry; then, if we take an axis of x to pass through M, the coordinates of M are (Ϛ, 0, 0). On the same system the coordinates of dS are a sin θcοsφ, a sin θ sin φ, α(l - cos ϴ) + b; and the distance is given by d 2 = b 2 + ζ 2 - 2α^sin 0 cos φ + 4«(a + b) sin 2 ⅛0.

In this expression ζ and θ are to be treated as small quantities. Writing p for a sin θ, we get approximately

, 7 χ 4 ∕ 2 <^ cos ≠ . , g + 0 2 d ~ b+ 2b~~b~ p+ 2⅛b p ( χ )∙

The vibration at the wave-front of resolution being denoted by a~ 1 cos 2ιrt∕τ, the integral expressive of the resultant of the secondary waves is (§17)

< 2 >∙

Substituting pdp dφ for rfS, and for d its value from (1), we obtain as the expression for the intensity at the point + < S >·

where

C =j∕∕'cos (¾κ∙p 2 - lpcosφ).pdpdφ, . . . . (4), 4 S=∕7^sin(⅜κp 2 -lp cos φ).pdpdφ . . . . (5), and the following abbreviations have been introduced 2-κ a + b 1 2τr< ’...

λ 2ab ⅛ κ ’ λb l (

The range of integration is for φ from 0 to 2π. The limits for p depend upon the particular problem in hand; but for the sake of definiteness we will suppose that in the analytical definitions of C and S the limits are 0 and r, so as to apply immediately to the problem of a circular aperture of radius r. If we introduce the notation of Bessel’s functions, we have = ifjp)cοs(fκp 2 ).pdp . . . . (7).

S = 2π^' J 0 (7p)sin (⅜κp 2 ).p√p .... (8).

By integration by parts of these expressions Lommcl develops series suitable for calculation. Setting

κr 2 = y, lr = z (9), he finds in the first place c ..,= j'⅛ l - ι+ ⅛ l - i ∣. .... (io), S = , r =j⅛U 1 - c ⅛U s ∣ .... (11 >, where u 1 =^j 1 ω-Sj 3 ω+Jj 5 ω- .... (12), u 2 -^j 2 ω-⅛∖ω+ (13).

These series are convenient when y is less than z.

The second set of expressions are r , „ ( 2 . z 2 ι sin⅜y v cos⅜y v ∕ .

c -^ιr m ⅛ + τr v ^^τr v, i · · () ·

(2 z 2 cos ⅛y yτ sin ⅜y 5r ) .

s -^k∞⅞^τr v ∙^τr ι s · · <15) ’

where

Vo = Jo(2)~^2 ∙L(2 )4'^4 J 4 (z ) - (lθ)j

V 1 =^j 1 ω-^ j 3 ω+ (π).

These series are suitable when z∣y is srnall.

When the primary wave is complete, r=∞, and we have at once from the second set of expressions n 2τr . l? α 2ιr Z 2 C “κ sin 2κ , S ” ∕c c ° s 2κ · · · · < 18 )’

so that

C 2 + S 2 1

T2_ v oo /ιοί

Cα⅜ 2 λ 2 ^(a + δ) 2 μy ' , as we know it should be.

In the application to the problem of the shadow of a circular disk the limits of integration are from r to ∞ . If these integrals be denoted by C', S', we have σ . fc . c .^∣.⅛ τ , t !⅛ τ ,j . . (20 ), ⅛-S.-S-^{^ιV β+ ^V 1 ∣. . . (21); and

C' 2 + S' 2 *^(V 0 2 + V 1 2 ) (22),

V 2 4- V 2 l·== (α + δ) 2 (23) ·

When the point where the illumination is required is situated upon the axis, £ l, z are zero. Hence V o = l,V 1 = 0, and

I 2 = -

(α + δ) 2 '

the same as if the primary wave had come on unbroken. This is Poisson’s theorem, already found (§ 10) by a much simpler method, in which attention is limited from the first to points upon the axis. The distribution of light at other points upon the screen is to be found from (23) by means of the series (16), (17) for V o and V 1 . Lommel gives curves for the intensity when y = π, 2π, 3π, . . . 6π. The bright central spot is accompanied by rings of varying intensity.

The limit of the geometrical shadow [£/(« +δ) = r∕α] corresponds to y=z. In this case

V 0 =J 0 (2)-J 2 (2 ) + J 4 (2 )- · · ∙=⅜{J0(2 ) + cosz } · · (24), V 1 =J 1 (z)-J 8 (z) + J 6 (z)- . . .=⅜sinz (25).

The numbers computed for special values of y and z apply to a whole class of problems. Since y= a + b a 2π∙ ζ ∙~κT* *-*Γ ’ > both y and z remain unchanged, even when λ is constant, if we suppose

δ«α, ?·oc£oc.\/a (26).

We may fall back upon Fraunhofer’s phenomena by supposing α = δ = 00, or more generally b = - a, so that y = 0.

Under these circumstances

C=πr 2 ½Φ, S = 0.

z

But it is unnecessary to add anything further under this head.

§ 19. Polarization.

A ray of ordinary light is symmetrical with respect to the direction of propagation. If, for example, this direction be vertical, there is nothing that can be said concerning the north and south sides of the ray that is not equally true concerning the east and west sides. In polarized light this symmetry is lost. Huygens showed that when a ray of such light falls upon a crystal of Iceland spar, which is made to revolve about the ray as an axis, the phenomena vary in a manner not to be represented as a mere revolution with the spar. In Newton’s language, the ray itself has sides, or is polarized.

Malus discovered that ordinary light may be polarized by reflexion as well as by double refraction; and Brewster proved that the effect is nearly complete when the tangent of the angle of incidence is equal to the refractive index, or (which comes to tho same) when the reflected and refracted rays are perpendicular to one another. The light thus obtained is said to be polarized in the plane of reflexion.

Reciprocally, the character of a polarized ray may be revealed by submitting it to the test of reflexion at the appropriate angle. As the normal to the reflecting surface revolves (in a cone) about the ray, there are two azimuths of the plane of incidence, distant 180°, at which the reflexion is a maximum, and two others, distant 90° from the former, at which the reflexion (nearly) vanishes. In the latter case the plane of incidence is perpendicular to that in which the light must be supposed to have been reflected in order to acquire its polarization.

The full statement of the law of double refraction is somewhat complicated, and scarcely to be made intelligible except in terms of the wave theory; but, in order merely to show the relation of double refraction in a uniaxal crystal, such as Iceland spar, to polarized light, we may take the case of a prism so cut that the refracting edge is parallel to the optic axis. By traversing such a prism, in a plane perpendicular to the edge, a ray of ordinary light is divided into two, of equal intensity, each of which is refracted according to the ordinary law of Snell. Whatever may be the angle and setting of the prism, the phenomenon may be represented by supposing half the light to be refracted with one index (1⋅65), and the other half with the different index (1⋅48). The rays thus arising are polarized,—the one more refracted in the plane of refraction, and the other in the perpendicular plane. If these rays are now allowed to fall upon a second similar prism, held so that its edge is parallel to that of the first prism, there is no further duplication. The ray first refracted with index 1 ’ 65 is refracted again in like manner, and similarly the ray first refracted with index 1⋅48 is again so refracted. But the case is altered if the second prism be caused to rotate about the incident ray. If the rotation be through an angle of 90°, each ray is indeed refracted singly; but the indices are exchanged. The ray that suffered most refraction at the first prism now suffers least at the second, and vice versa. At intermediate rotations the double refraction reasserts itself, each ray being divided into two, refracted with the above-mentioned indices, and of intensity dependent upon the amount of rotation, but always such that no light is lost (or gained) on the whole by the separation.

The law governing the intensity was formulated by Malus, and has been verified by the measures of Arago and other workers. If θ be the angle of rotation from the position in which one of the rays is at a maximum, while the other vanishes, the intensities are proportional to cos 2 ϴ and sin 2 ϴ . On the same scale, if we neglect the loss by reflexion and absorption, the intensity of the incident light is represented by unity.

A similar law applies to the intensity with which a polarized ray is reflected from a glass surface at the Brewsterian angle. If θ be reckoned from the azimuth of maximum reflexion, the intensity at other angles may be represented by cos 2 ϴ , vanishing when ϴ = 90°.

The phenomena here briefly sketched force upon us the view that the vibrations of light are transverse to the direction of propagation. In ordinary light the vibrations are as much in one transverse direction as in another; and when such light falls upon a doubly refracting, or reflecting, medium, the vibrations are resolved into two definite directions, constituting two rays polarized in perpendicular planes, and differently influenced by the medium. In this case the two rays are necessarily of equal intensity.

Consider, for example, the application of this idea to the reflexion of a ray of ordinary light at the Brewsterian, or polarizing, angle. The incident light may be resolved into two, of equal intensity, and polarized respectively in and perpendicular to the plane of incidence. Now we know that a ray polarized in the plane perpendicular to that of incidence will not be reflected, will in fact be entirely transmitted; aud the necessary consequence is that all the light reflected at this angle will be polarized in the plane of incidence. The operation of the plate is thus purely selective, the polarized component, which is missing in the reflected light, being represented in undue proportion in the transmitted light.

If the incident light be polarized, suppose at an angle ϴ with the plane of incidence, the incident vibration may be resolved into cos ϴ in the one plane and sin ϴ in the other. The latter polarized component is not reflected. The reflected light is thus in all cases polarized in the plane of reflexion; and its intensity, proportional to the square of the vibration, is represented by h cos 2 θ , if h be the intensity in which light is reflected when polarized in the plane of reflexion. The law of Malus is thus a necessary consequence of the principle of resolution.

The idea of transverse vibrations was admitted with reluctance, even by Young and Fresnel themselves. A perfect fluid, such as the ethereal medium was then supposed to be, is essentially incapable of transverse vibrations. But there seems to be no reason a priori for preferring one kind of vibration to another; and the phenomena of polarization prove conclusively that, if luminous vibrations are analogous to those of a material medium, it is to solids, and not to fluids, that we must look. An isotropic solid is capable of propagating two distinct kinds of waves,—the first dependent upon rigidity, or the force by which shear is resisted, and the second analogous to waves of sound aud dependent upon compressibility. In the former the vibrations are transverse to the direction of propagation, that is, they may take place in any direction parallel to the wave front, and they are thus suitable representatives of the vibrations of light. In this theory the luminiferous ether is distinctly assimilated to an elastic solid, and the velocity of light depends upon the rigidity and density assigned to the medium.

The possibility of longitudinal waves, in which the displacement is perpendicular to the wave-front, is an objection to the elastic solid theory of light, for there is nothing known in optics corresponding thereto. If, however, we suppose with Green that the medium is incompressible, the velocity of longitudinal waves becomes infinite, and the objection is in great degree obviated. Such a supposition is hardly a departure from the original idea, inasmuch as, so far as we know, there is nothing to prevent a solid material possessing these properties, and an approximation is actually presented by such bodies as jelly, for which the velocity of longitudinal vibrations is a large multiple of that of transverse vibrations.

§ 20. Interference of Polarized Light.

The conditions of interference of polarized light are most easily deduced from the phenomena of the colours of crystalline plates, if we once admit Young’s view that the origin of the colours is to be sought in the interference of the differently refracted rays. Independently of any hypothesis of this kind, the subject was directly investigated by Fresnel and Arago, who summarized their conclusions thus :—

(1) Under the same conditions in which two rays of ordinary light appear to destroy one another, two rays polarized in contrary (viz., perpendicular) directions are without mutual influence.

(2) Two rays of light polarized in the same direction act upon one another like ordinary rays; so that, with these two kinds of light, the phenomena of interference are identical.

(3) Two rays originally polarized in opposite directions may afterwards be brought to the same plane of polarization, without thereby acquiring the power to influence one another.

(4) Two rays polarized in opposite directions, and afterwards brought to similar polarizations, react in the same manner as natural rays, if they are derived from a beam originally polarized in one direction,.

The fact that oppositely polarized rays cannot be made to interfere may of itself be regarded as a proof that the vibrations are transverse; and the principle, once admitted, gives an intelligible account of all the varied phenomena in this field of optics. The only points on which any difficulty arises are as to the nature of ordinary unpolarized light, and the rules according to which intensity is to be calculated. It will be proper to consider these questions somewhat fully.

In ordinary (plane) polarized light the vibrations are supposed to be in one direction only. If x and y be rectangular coordinates in the plane of the wave, we may take, as representing a regular vibration of plane-polarized light, x=acos(ϕ-α ) (1), where φ = 2πt∕τ , and a, a denote constants. It must be remembered, however, that in optics a regular vibration of this kind never presents itself. In the simplest case of approximately monochromatic light, the amplitude and phase must be regarded (§ 4) as liable to incessant variation, and all that we are able to appreciate is the mean intensity, represented by M(a 2 ). If a number of these irregular streams of light are combined, the intensity of the mixture cannot be calculated from a mere knowledge of the separate intensities, unless we have assurance that the streams are independent, that is, without mutual phase-relations of a durable character. For instance, two thoroughly similar streams combine into one of fourfold intensity, if the phases are the same; while, if the phases are opposed, the intensity falls to zero. It is only when the streams are independent, so that the phase-relation is arbitrary and variable from moment to moment, that the apparent resultant intensity is necessarily the double of the separate intensities.

If any number of independent vibrations of type (1) be superposed, the resultant is

[∑ α 1 cos α 1 3 cos φ + [Σ a 1 sin α 1 ] sin φ, and the momentary intensity is [∑ a 1 cos α 1 ] 2 + [Σ a 1 sin a 1 ] 2 , or a ι 2 + i ⅛ 2 + · · · + 2a 1 a 2 cos (a 1 - a. 2 ) +. ..

The phase-relations being unknown, this quantity is quite indeterminate. But, since each cosine varies from moment to moment, and on the whole is as much positive as negative, the mean intensity is M(α 1 2 ) + M(α a 2 )+ ..., that is to say, is to be found by simple addition of the separate intensities.

Let us now dispense with the restriction to one direction of vibration, and consider in the first place the character of a regular vibration, of given frequency. The general expression will be x =αcos(φ - a), y=bcοs(ϕ-β) . . . (2), where a, a, b, β are constants. If β = a, the vibrations are executed entirely in the plane x∣y=a∣b, or the light is plane-polarized. Or if β = π - a, the light is again plane-polarized, the plane of vibration being x∣y= -a∣b. In other cases the vibrations are not confined to one plane, so that the light is not plane-polarized, but, in conformity with the path denoted by (2), it is said to be elliptically- polarized. If one of the constituents of elliptically-polarized light be suitably accelerated or retarded relatively to the other, it may be converted into plane-polarized light, and so identified by the usual tests. Or, conversely, plane-polarized light may be converted into elliptically-polarized by a similar operation. The relative acceleration in question is readily effected by a plate of doubly refracting crystal cut parallel to the axis.

If β = α±½π, whether in the first instance or after the action of a crystalline plate, x = αcos(ϕ-α), y = ± b sin (ϕ-α ) .. . (3).

The maxima and minima values of the one coordinate here occur synchronously with the evanescence of the other, and the coordinate axes are the principal axes of the elliptic path.

An important particular case arises when further δ = α, The path is then a circle, and the light is said to be circularly -polarized. According to the sign adopted in the second equation (3), the circle is described in the one direction or in the other.

Circularly-polarized light can be resolved into plane-polarized components in any two rectangular directions, which are such that the intensities are equal and the phases different by a quarter period. If a crystalline plate be of such thickness that it retards one component by a quarter of a wave-length (or indeed by any odd multiple thereof) relatively to the other, it will convert plane-polarized light into circularly-polarized, and conversely,—in the latter case without regard to the azimuth in which it is held.

The property of circularly-polarized light whereby it is capable of resolution into oppositely plane-polarized components of equal intensities is possessed also by natural unpolarized light; but the discrimination may be effected experimentally with the aid of the quarter-wave plate. By this agency the circularly-polarized ray is converted into plane-polarized, while the natural light remains apparently unaltered. The difficulty which remains is rather to explain the physical character of natural light. To this we shall presently return; but in the meantime it is obvious that the constitution of natural light is essentially irregular, for we have seen that absolutely regular, i . e., absolutely homogeneous, light is necessarily (elliptically) polarized.

In discussing the vibration represented by (2), we have considered the amplitudes and phases to be constant; but in nature this is no more attainable than in the case of plane-polarized light. In order that the elliptic polarization may be of a definite character, it is only necessary that the ratio of amplitudes and the difference of phases should be absolute constants, and this of course is consistent with the same degree of irregularity as was admitted for plane vibrations.

The intensity of elliptically-polarized light is the sum of the intensities of its rectangular components. This we may consider to be an experimental fact, as well as a consequence of the theory of transverse vibrations. In whatever form such a theory may be adopted, the energy propagated will certainly conform to this law. When the constants in (2) are regarded as subject to variation, the apparent intensity is represented by

M(α 2 ) + M(b 2 ) (4).

We are now in a position to examine the constitution which must be ascribed to natural light. The conditions to be satisfied are that when resolved in any plane the mean intensity of tho vibrations shall be independent of the orientation of the plane, and, further, that this property shall be unaffected by any previous relative retardation of the rectangular components into which it may have been resolved. The original vibration being represented by x =αcos(ϕ-α), y = b cos(φ - β), or, as we may write it, since we are concerned only with phase differences, x =αcosϕ, y=bcos(ϕ-δ ) . . . (5), let us suppose that the second component is subjected to a retardation ϵ. Thus x=acosφ, y=bcοs(φ-δ -ϵ) . . . (6), in which a, b, δ will be regarded as subject to rapid variation, while ϵ remains constant. If the vibration represented by (6) be now resolved in a direction x', making an angle ω with x, we have x! — a cos φ cos ω + b cos (<∕> - δ - e) sin ω = [a cos ω + b sin ω cos (δ + e)] cos φ + b sin ω sin (δ + e) sin φ; and the intensity is

α 2 cos 2 ω + δ 2 sin 2 ω + 2αδ cos ω sin « cos (δ + ∈) . . (7).

Of this expression we take the mean, ω and e remaining constant. Thus the apparent intensity may be written M(z' 2 ) = M(α 2 ) cos 2 « + M(δ 2 ) sin 2 ω + 2M[αδ cos (δ + ∈)] cos ω sin ω (8). In order now that the stream may satisfy the conditions laid down as necessary for natural light, (8) must be independent of ω and ∈; so that

M(α 2 )-M(δ 2 ) (9),

ΛI(αδcos δ) = M(αδsin δ) = 0 (10).

In these equations a 2 and δ 2 represent simply the intensities, or squares of amplitudes, of the x and y vibrations; and the other two quantities admit also of a simple interpretation. The value of y may bo written

τ∕=δcosδcos<∕> + δsinδsin<∕> .... (ll); from which we see that δcosδ is the coefficient of that part of the y vibration which has the same phase as the x vibration. Thus αδcosδ may be interpreted as the product of the coefficients of the parts of the x and y vibrations which have the same phase. Next suppose the phase of y accelerated by writing ⅜π + <∕> in place of φ. We should thus have y = -δcosδsin≠ + δsinδcos≠, andαδsinδ represents the product of the coefficients of the parts which are now in the same phase, or (which is the same) the product of the coefficients of the x vibration and of that part of the y vibration which was 90° behind in phase. In general, if x≈hcosφ + h'sinφ, y≈k cosφ+k , sin≠ . (12), the first product is hk + h'P and the second is hV-Bk.

Let us next examine how the quantities which we have been considering are affected by a transformation of coordinates in accordance with the formulae a∕=2cosω + wsin ω , √=-aesin «-H/cos « . (13).

We find a∕ = cos ≠{αcos ω + δsin ω cos δ} +sin φ . δ sin δ sin ω . (14), y' = cos√>{ - αsinω + δcosωcosδ} +sin φ. b sin δ cos ω . (15); whence amp. 2 of23 ’ = α 2 cos 2 ω + δ 2 sin 2 ω + 2αδcosδsinωcosω . (16), amp. 2 of ?/'= α 2 sin 2 « + δ 2 cos 2 « - 2αδ cos δ sin « cos « . (17).

In like manner

First product = (δ 2 - α 2 ) sin ω cos u> + ab cos δ (cos 2 « - sin 2 ω) (18), Second product =ab sin δ (19).

The second product, representing the circulating part of the motion, is thus unaltered by the transformation.

Let us pass on to the consideration of the mean quantities which occur in (9), (10), writing for brevity

M(α 2 ) = A, M(δ 2 ) = B, M(αδcosδ) = C, M(αδsinδ) = D.

From (16), (17), (18), (19), if A', B', C', D' denote the corresponding quantities after transformation,

A' = A cos 2 ω + B sin 2 ω + 2C cos « sin ω .... (20),

B' = A sin 2 « + B cos 2 « - 2C cos ω sin ω . . . . (21),

C' = C(cos 2 ω - sin 2 ω) + (B - A) cos ω sin ω . . . (22),

D' = D (23).

These formulae prove that, if the conditions (9), (10), shown to be necessary in order that the light may behave as natural light, be satisfied for one set of axes, they are equally satisfied with any other. It is thus a matter of indifference with respect to what axes the retardation e is supposed to be introduced, and the conditions (9), (10) are sufficient, as well as necessary, to characterize natural light.

Reverting to (8), we see that, whether the light be natural or not, its character, so far as experimental tests can show, is determined by the values of A, B, C, D. The effect of a change of axes is given by (20), &c., and it is evident that the new axes may always be so chosen that C'=0. For this purpose it is only necessary to take ω such that tan 2ω = 2C∕(A-B).

If we choose these new axes as fundamental axes, the values of the constants for any others inclined to them at angle ω will be of the form

A = A 1 cos 2 ω + B 1 sin 2 ω }

B = A 1 sin 2 ω+B 1 cos 2 α> > (24).

C = (B 1 - A 1 ) cos ω sin ω )

If A 1 and B 1 are here equal, then C∙=0, A = B for all values of ω. In this case, the light cannot be distinguished from natural light by mere resolution; but if D be finite, the difference may be made apparent with the aid of a retarding plate.

If A 1 and B 1 are unequal, they represent the maximum and minimum values of A and B. The intensity is then a function of the plane of resolution, and the light may be recognized as partially polarized by the usual tests. If either A 1 or B 1 vanishes, the light is plane-polarized.

When several independent streams of light are combined, the values, not only of A and B, but also of C and D, for the mixture, are f to be found by simple addition. It must here be distinctly understood that there are no permanent phase-relations between one component and another. Suppose, for example, that there are two streams of light, each of which satisfies the relations A = B, C≈=0, but makes the value of D finite. If the two values of D are equal and opposite, and the streams are independent, the mixture constitutes natural light. A particular case arises when each component is circularly-polarized (D=±A=±B), one in the right-handed and the other in the left-handed direction. The intensities being equal, the mixture is equivalent to natural light, but only under the restriction that the streams are without phase-relation. If, on the contrary, the second stream be similar to the first, affected merely with a constant retardation, the resultant is not natural, but completely (plane) polarized light.

We will now prove that the most general mixture of light may be regarded as compounded of one stream of light elliptically-polarized in a definite manner, and of an independent stream of natural light. The theorem is due to Stokes, but the method that we shall follow is that of Verdet.

In the first place, it is necessary to observe that the values of the fundamental quantities A, B, C, D are not free from restriction. It will be shown that in no case can C 2 + D 2 exceed AB.

In equations (2), expressing the vibration at any moment, let α ι> ⅛ 0t ι> 3 υ be the values of a, b, a, β during an interval of time proportional to rn 1 , and in like manner let the suffixes 2, 3, . . . . correspond to times proportional to nι 2 , m 3 , .... Then

AB=wι 1 2 α 1 2 δ 1 2 +m 2 2 α 2 2 δ 2 2 +. .. + m 1 rn 2 (α 1 2 δ 2 2 + α 2 2 δ, 2 ) + . .. Again, by (12),

C=τn 1 α 1 δ 1 (cos α 1 cos β 1 + sin a 1 sin 0 1 ) + ...

= wι 1 a 1 δ 1 cos δ 1 + 7n 2 a 2 δ 2 cos δ 2 +... ,

D =rn 1 a 1 b 1 sin δ 1 + w⅛ a 2 δ 2 sin δ 2 +. . .; where, as before,

δι = 3ι - b2~β∙2~ a 2> ∙∙∙∙

Thus,

C 2 + D 2 =w 1 2 α 1 2 δ 1 2 + 7n 2 2 α 2 2 δ 2 2 +.. . + rn 1 m 2 a 1 b 1 a 2 b 2 cos (δ 2 - δ 1 ) +...

From these equations we see that AB - C 2 - D 2 reduces itself to a sum of terms of the form m 1 vι 2 [a 1 2 b 2 2 + α 2 2 δ 1 2 - 2α 1 δ 1 α 2 δ 2 cos (δ 2 - δ 1 )], each of which is essentially positive.

The only case in which the sum can vanish is when

δj = δ 2 = δ 2 =. . . , and further δ 1 : a j = δ 2 : a 2 = δ 3 : a 3 =. ..

Under these conditions the light is reduced to be of a definite elliptic character, although the amplitude and phase of the system as a whole may be subject to rapid variation. The elliptic constants are given by

δ 2 ∕α 2 =B∕A, tanδ = D∕C. . . . (25).

In general AB exceeds (C 2 + D 2 ); but it will always be possible to find a positive quantity H, which when subtracted from A and B (themselves necessarily positive) shall reduce the product to equality with C 2 + D 2 , in accordance with

(A-H)(B-H) = C 2 + D 2 (26).

The original light may thus be resolved into two groups. For the first group the constants are H, H, 0, 0; and for the second A - H, B - H, C, D. Each of these is of a simple character; for the first represents natural light, and the second light elliptically polarized. It is thus proved that in general a stream of light may be regarded as composed of one stream of natural light and of another elliptically-polarized. The intensity of the natural light is 2H, where from (26)

H=⅛(A + B)-⅜√{(A-B) 2 + 4(C 2 + D 2 )} . . (27).

The elliptic constants of the second component are given by

δ 2 ∕α 2 =(B - H)∕(A-H), tanδ = D∕C . . (28), and

M(α 2 ) = A-H ....... (29).

If D = 0, and therefore by (28) δ = 0, the second component is plane-polarized. This is regarded as a particular case of elliptic polarization. Again, if A = B, C = 0, the polarization is circular.

The laws of interference of polarized light, discovered by Fresnel and Arago, are exactly what the theory of transverse vibrations would lead us to expect, when once we have cleared up the idea of unpolarized light. Ordinary sources, such as the sun, emit unpolarized light. If this be resolved in two opposite directions, the polarized components are not only each irregular, but there is no permanent phase-relation between them. No light derived from one can therefore ever interfere regularly with light derived from the other. If, however, we commence with plane-polarized light, we have only one series of irregularities to deal with. When resolved in two rectangular directions, the components cannot then interfere, but only on account of the perpendicularity. If brought back by resolution to the same plane of polarization, interference becomes possible, because the same series of irregularities are to be found in both components.

§ 21. Double Refraction.

The construction by which Huygens explained the ordinary and extraordinary refraction of Iceland spar has already been given (Light, vol. xiv. p. 610). The wave-surface is in two sheets, composed of a sphere and of an ellipsoid of revolution, in contact with one another at the extremities of the polar axis. In biaxal crystals the wave-surface is of a more complicated character, including that of Huygens as a particular case.

It is not unimportant to remark that the essential problem of double refraction is to determine the two velocities with which plane waves are propagated, when the direction of the normal to the wave-front is assigned. When this problem has been solved, the determination of the wave-surface is a mere matter of geometry, not absolutely necessary for the explanation of the leading phenomena, but convenient as affording a concise summary of the principal laws. In all cases the wave-surface is to be regarded as the envelope at any subsequent time of all the plane wave-fronts which at a given instant may be supposed to be passing through a particular point.

In singly refracting media, where the velocity of a wave is the same in all directions, the wave-normal coincides with the ray. In doubly refracting crystals this law no longer holds good. The principles by which the conception of a ray is justified (§ 10), when applied to this case, show that the centre of the zone system is not in general to be found at the foot of the perpendicular upon the primary wave-front. The surface whose contact with the primary wave-front determines the element from which the secondary disturbance arrives with least retardation is now not a sphere, but whatever wave-surface is appropriate to the medium. The direction of the ray, corresponding to any tangent plane of the wavesurface, is thus not the normal, but the radius vector drawn from the centre to the point of contact.

The velocity of propagation (reckoned always perpendicularly to the wave-front) may be conceived to depend upon the direction of the wave-front, or wave-normal, and upon what we may call (at any rate figuratively) the direction of vibration. If the velocity depended exclusively upon the wave-normal, there could be no double, though there might be extraordinary, refraction, i.e., refraction deviating from the law of Snell; but of this nothing is known in nature. The fact that there are in general two velocities for one wave-front proves that the velocity depends upon the direction of vibration.

According to the Huygenian law, confirmed to a high degree of accuracy by the observations of Brewster and Swan, a ray polarized in a principal plane (i.e ., a plane passing through the axis) of a uniaxal crystal suffers ordinary refraction only, that is, propagates itself with the same velocity in all directions. The interpretation which Fresnel put upon this is that the vibrations (understood now in a literal sense) are perpendicular to the plane of polarization, and that the velocity is constant because the direction of vibration is in all cases similarly related (perpendicular) to the axis. The development of this idea in the fertile brain of Fresnel led him to the remarkable discovery of the law of refraction in biaxal crystals.

The hypotheses upon which Fresnel based his attempt at a mechanical theory are thus summarized by Verdet:—

(1) The vibrations of polarized light are perpendicular to the plane of polarization;

(2)The elastic forces called into play during the propagation of a system of plane waves (of rectilinear transverse vibrations) differ from the elastic forces developed by the parallel displacement of a single molecule only by a constant factor, independent of the particular direction of the plane of the wave;

(3) When a plane wave propagates itself in any homogeneous medium, the components parallel to the wave-front of the elastic forces called into play by the vibrations of the wave are alone operative;

(4) The velocity of a plane wave which propagates itself with type unchanged in any homogeneous medium is proportional to the square root of the effective component of the elastic force developed by the vibrations.

Fresnel himself was perfectly aware that his theory was deficient in rigour, and indeed there is little to be said in defence of his second hypothesis. Nevertheless, the great historical interest of this theory, and the support that experiment gives to Fresnel’s conclusion as to the actual form of the wave-surface in biaxal crystals, render some account of his work in this field imperative.

The potential energy of displacement of a single molecule from its position of equilibrium is ultimately a quadratic function of the three components reckoned parallel to any set of rectangular axes. These axes may be so chosen as to reduce the quadratic function to a sum of squares, so that the energy may be expressed,

V = ⅜α 2 ξ 2 + W + ⅜cT (1), where ξ, η, Ϛ are the three component displacements. The corresponding forces of restitution, obtained at once by differentiation, are X-α 2 ξ, Y = Z⅜, Z=c 2 < . . . (2).

The force of restitution is thus in general inclined to the direction of displacement. The relation between the two directions X, Y, Z and ξ, η, Ϛ is the same as that between the normal to a tangent plane aud the radius vector p to the point of contact in the ellipsoid α 2 ξ 2 + δV + c 2 f 2 ≈l (3).

If a 2 , b 2 , c 2 are unequal, the directions of the coordinate axes are the only ones in which a displacement calls iuto operation a parallel force of restitution. If two of the quantities a 2 , b 2 , c 2 are equal, the ellipsoid (3) is of revolution, and every direction in the plane of the equal axes possesses the property in question. This is the case of a uniaxal crystal. If the three quantities a 2 , b 2 , c 2 are all equal, the medium is isotropic.

If we resolve the force of restitution in the direction of displacement, we obtain a quantity dependent upon this direction in a manner readily expressible by means of the ellipsoid of elasticity (3). For, when the total displacement is given, this quantity is proportional to α⅞ 2 + b 2 p 2 + c 2 ζ 2 ζ 2 + η 2 +ζ 2 ’ that is to say, to the inverse square of the radius vector p in (3).

We have now to inquire in what directions, limited to a particular plane, a displacement may be so made that the projection of the force of restitution upon the plane may be parallel to the displacement. The answer follows at once from the property of the ellipsoid of elasticity. For, if in any section of the ellipsoid we have a radius vector such that the plane containing it and the normal to the corresponding tangent plane is perpendicular to the plane of the section, the tangent line to the section must be perpendicular to the radius vector, that is, the radius vector must be a principal axis of the section. There are therefore two, and in general only two, directions in any plane satisfying the proposed condition, and these are perpendicular to one another. If, however, the plane be one of those of circular section, every line of displacement is such that the component of the force, resolved parallel to the plane, coincides with it.

According to the principles laid down by Fresnel, we have now complete data for the solution of the problem of double refraction. If the direction of the wave-front be given, there are (in general) only two directions of vibration such that a single wave is propagated. If the actual displacements do not conform to this condition, they will be resolved into two of the required character, and the components will in general be propagated with different velocities. The two directions are the principal axes of the section of (3) made by the wave-front, and the velocities of propagation are inversely proportional to the lengths of these axes.

The law connecting the lengths of the axes with the direction (l, m, n) of the plane is a question of geometry; and indeed the whole investigation of the wave-surface may be elegantly carried through geometrically with the aid of certain theorems of MacCullagh respecting apsidal surfaces (Salmon, ch. xiv.). For this, however, we have not space, and must content ourselves with a sketch of the analytical method of treatment.

If v be the velocity of propagation in direction l, m, n, the wavesurface is the envelope of planes lx+my + nz=υ (4), where v is a function of I, m, n, whose form is to be determined. If (λ, μ, v) be the corresponding direction of vibration, then lλ + mμ+nv=0 (5).

According to the principles laid down by Fresnel, we see at once that the force of restitution (a 2 λ, b 2 μ, c 2 v), corresponding to a displacement unity, is equivalent to a force v 2 along (λ, μ, v), together with some force (P) along (l , m, n). Resolving parallel to the coordinate axes, we get

ZP=α 2 λ-v 2 λ, mP = b 2 μ-v 2 μ, riP=c 2 r-v 2 v, or

_ ZP _ ?nP _ 7iP .

λ ~ft 2 → 2 ' "~C^ 2 → 2 ' ‘ '

Multiplying these by l, m, n respectively, and taking account of (5) , we see that

Z 2 m 2 n2 (>7∖

2+^2Z^2- υ Oi is the relation sought for between υ and (l , m, n). In this equation b, c are the velocities when the direction of propagation is along x, the former being applicable when the vibration is parallel to y , and the latter when it is parallel to x.

The directions of vibration are determined by (5) and by the consideration that (l , m, n), (λ, μ, v ), and (α 2 λ, b 2 μ, c 2 v ) lie in a plane, or (as we may put it) are all perpendicular to one direction (f, g, h).

Thus lf+ mg + nh ≈0)

λ∕+ μ<7+*'⅛ =θ r (θ)∙

α 2 λ∕ + b 2 μg + <?vh= 0 J

The determinant expressing the result of the elimination of/: g: h may be put into the form

⅛. c 2∫ + ⅛- α 2) + ⅛-δ 2 )-0. . . . (9),

λ v M v which with (5) suffices to determine (λ, μ, v) as a function of (Z, m, n).

The fact that the system of equations (5), (8) is symmetrical as between (λ, μ, v) and (∕, g, h~) proves that the two directions of vibration corresponding to a given (Z, m, ri) are perpendicular to one another.

The direct investigation of the wave-surface from (4) and (7) was first effected by Ampère, but his analytical process was very laborious. Fresnel had indeed been forced to content himself with an indirect method of verification. But in the following investigation of A. Smith the eliminations are effected with comparatively little trouble.

In addition to (4) and (7), we know that

Z 2 + m 2 + n 2 = 1 (10).

To find the equation to the envelope, we have to differentiate these equations, making l, m, n, υ vary. Eliminating the differentials by the method of multipliers, we obtain the following :—

ae=AZ +BZ∕(√ 2 -α 2 ) (ll), y = Ara, + Bnι∕(υ 2 - δ 2 ) (12), z=An + Bn∕('υ 2 -c 2 ) (13); and 1=Bv j (v 2 -α 2 ) 2+ (^-δ 2 ) 2+ (^-c 2 ) 2 } · · , (14) ·

The equations (ll), (12), (13) multiplied by l, m, n respectively and added, give υ=A ........ (15).

The same equations, squared and added, give ac 2 +y 2 + 2 2 =A 2 + B/v.

If we put r 2 for x i + y 2 + z 2 , and for A the value just found, we obtain B = ∙υ(r 2 --y 2 ) (16).

If these values of A and B be substituted in (ll), 7 ( 7 r 2 - v 2 ) 7 r 2 - α 2 x=lυ < 1 + -5 5 > =lv -0 s, ( ri - a t ) ri-a 2 or

*=⅛ ξ ⅛- (17 )∙

ria i υ

If we substitute this value of l, and the corresponding values of m, n in (4), we get (v 2 -a 2 )x 2 (v 2 -δ 2 )y 2 (ri - c 2 ) z 2 _ 2 _ riri riy 2 riz 2 ri - a 2 r 2 - b 2 r 2 -ri V r a ÷ r 2 r 2 > whence

≠⅛ + ≠ς + ∕⅛. 0 as), ri — a i r~ — ri ri — c as the equation of the wave-surface.

By (6) equation (ll) may be written x=AZ+BP -1 λ, from which and the corresponding equations we see that the direction (x, y, z) lies in the same plane as (Z, m, ri) and (λ, μ, v). Hence in any tangent plane of the wave-surface the direction of vibration is that of the line joining the foot of the perpendicular and the point of contact (x, y, z).

The equation (18) leads to another geometrical definition of Fresnel’s wave-surface. If through the centre of the ellipsoid reciprocal to the ellipsoid of elasticity (3), viz., x 2 ∕a 2 + y 2 ∕b 2 + z 2 ∣c 2 =l (19), a plane be drawn, and on the normal to this plane two lengths be marked off proportional to the axes of the elliptic section determined by the plane, the locus of the points thus obtained, the apsidal surface of (19), is the wave-surface (18).

Fully developed in integral powers of the coordinates, (18) takes the form

(x 2 + y 2 + z 2 )(α⅛ 2 + b 2 y 2 + c 2 z 2 ) - α 2 (δ 2 + ri)x i

-δ 2 (c 2 + α 2 )y 2 -c 2 (α 2 + δ 2 )z 2 + α 2 δ 2 c 2 =0 . . . (20).

The section of (20) by the coordinate plane y=0 is

(ae 2 + z 2 - δ 2 )(α⅛ 2 + riz 2 — a 2 ri) = 0 .... (21), representing a circle and an ellipse (fig. 24). That the sections by each of the principal planes would be a circle and an ellipse might have been foreseen independently of a general solution of the envelope problem. The forms of the sections prescribed in (21) and the two similar equations are sufficient to determine the character of the wave-surface, if we assume that it is of the fourth degree, and involves only the even powers of the coordinates. It was somewhat in this way that the equation was first obtained by Fresnel.

If two of the principal velocities, e.g., a and δ, are equal, (20) becomes

(ae 2 + y 2 +z 2 - α 2 )(α¾ 2 + a 2 y 2

+ c⅜ 2 - α 2 c 2 ) = 0 . (22), so that the wave-surface degenerates into the Huygenian sphere and ellipsoid of revolution appropriate to a uniaxal crystal. The two sheets touch one another at the points x=0, y = 0, z=±α. If oa, as in Iceland spar, the ellipsoid is external to the sphere. On the other hand, if c<a, as in quartz, the ellipsoid is internal.

We have seen that when the wave-front is parallel to the circular sections of (3), the two wave velocities coincide. Thus in (7), if a 2 , b 2 , c 2 be in descending order of magnitude, we have m=0, v=b; so that w,a 1 /03)

a 2 - δ 2 δ 2 - c 2 α 2 - c 2

In general, if θ, θ' be the angles which the normal to the actual wave-front makes with the optic axes, it may be proved that the difference of the squares of the two roots of (7) is given by

¾ 2 - ¾ 2 = (α 2 - c 2 ) sin 0 sin 0' .... (24). In a uniaxal crystal the optic axes coincide with the axis of symmetry, and there is no distinction between 0 ' and 0.

Since waves in a biaxal crystal propagated along either optic axis have but one velocity, it follows that tangent planes to the wavesurface, perpendicular to these directions, touch both sheets of the surface. It may be proved further that each plane touches the surface not merely at two, but at an infinite number of points, which lie upon a circle.

The directions of the optic axes, and the angle included between them, are found frequently to vary with the colour of the light. Such a variation is to be expected, in view of dispersion, which renders α 2 , δ 2 , c 2 functions of the wave-length.

A knowledge of the form of the wave-surface determines in all cases the law of refraction according to the construction of Huygens. We will suppose for simplicity that the first medium is air, and that the surface of separation between the media is plane. The incident wave-front at any moment of time cuts the surface of separation in a straight line. On this line take any point, and with it as centre construct the wave-surface in the second medium corresponding to a certain interval of time. At the end of this interval the trace of the incident wave-front upon the surface will have advanced to a new position, parallel to the former. Planes drawn through this line so as to touch the wavesurface give the positions of the refracted wave-fronts. None other could satisfy the two conditions—(l)that the refracted wavefront should move within the crystal with the normal velocity suitable to its direction, and (2) that the traces of the incident and refracted waves upon the surface of separation should move together. The normal to a refracted wave lies necessarily in the plane of incidence, but the refracted ray, coinciding with the radius vector of the wave-surface, in general deviates from it. In most cases it is sufficient to attend to the wave normal.

As in total reflexion by simply refracting - media, it may happen that no tangent planes can be drawn to satisfy the prescribed conditions, or that but one such can be drawn.

When the crystal is uniaxal, one wave is refracted according to the ordinary law of Snell. The accuracy of both the sphere and the ellipsoid of the Huygenian construction has been fully verified by modern observations.

The simplest case of uniaxal refraction is when the axis of the crystal is perpendicular to the plane of incidence, with respect to which every thing then becomes symmetrical. The section of the wave-surface with which we have to deal reduces to two concentric circles; so that both waves are refracted according to the ordinary law, though of course with different indices.

In biaxal crystals one wave follows the ordinary law of refraction, if the plane of incidence coincide with a principal plane of the crystal. This consequence of his theory was verified by Fresnel himself, and subsequently by Rudberg and others. But the most remarkable phenomena of biaxal refraction are undoubtedly those discovered by Hamilton and Lloyd, generally known as conical refraction.

In general there are two refracted rays, corresponding to two distinct waves. But the refracted waves coalesce when they are perpendicular to either optic axis, and (as we have seen) this wave touches the wave-surface along a circle. Thus corresponding to one wave direction there are an infinite number of rays, lying upon a cone. The division of a single incident ray into a cone of refracted rays is called internal conical refraction. If the second face of the crystal is parallel to the first, each refracted ray resumes on emergence its original direction, so that the emergent bundle forms a hollow cylinder.

External conical refraction depends upon the singular points in the principal plane of zx, where the two sheets of the surface cross one another (fig. 24). At such a point (P) an infinite number of tangent planes may be drawn to the surface, and each of the perpendiculars from O represents a wave direction, corresponding to the single ray OP. On emergence these waves will be differently refracted; and thus corresponding to a single internal ray there are an infinite number of external rays, lying upon a cone.

It has already been admitted that the dynamical foundations of Fresnel’s theory are unsound; and it must be added that the rigorous theory of crystalline solids investigated by Cauchy and Green does not readily lend itself to the explanation of Fresnel’s laws of double refraction. On this subject the reader should consult Prof. Stokes’s Report. Sir W. Thomson has recently shown that an originally isotropic medium, pressed unequally in different directions, may be so constituted as to vibrate in accordance with Fresnel’s laws.

It may perhaps be worth while to remark that the equations, analogous to (2) § (24), which lead to these laws are

S = . . (25 ), where α, b, c are the principal wave velocities. If we here assume < = λ0, p≈μθ, ζ=*vθ, θ∕θ 0 = P/Po = c ik ^ lx + m ^+ nz - v '>, and substitute in (25), the condition of transversality leads at once to the desired results. But the equations (25) are not applicable to the vibrations of a crystalline solid.

In the electromagnetic theory double refraction is attributed to aeolotropic inductive capacity, and appears to offer no particular difficulty.

If the present position of the theory of double refraction is still somewhat unsatisfactory, it must be remembered that the uncertainty does not affect the general principle. Almost any form of wave theory involving transverse vibrations will explain the leading phenomenon, viz., the bifurcation of the ray. It is safe to predict that when ordinary refraction is well understood there will be little further trouble over double refraction.

The wave-velocity is not the only property of light rendered unsymmetrical by crystalline structure. In many cases the two polarized rays are subject to a different rate of absorption. Tourmalines and other crystals may be prepared in plates of such thickness that one ray is sensibly stopped and the other sensibly transmitted, and will then serve as polarizing (or analysing) apparatus. Although for practical purposes Nicol’s prisms (Light, vol. xiv. p. 612) are usually to be preferred, the phenomenon of double absorption is of great theoretical interest. The explanation is doubtless closely connected with that of double refraction.

§ 22. Colours of Crystalline Plates.

When polarized light is transmitted through a moderately thin plate of doubly refracting crystal, and is then analysed, e.g., with a Nicol, brilliant colours are often exhibited, analogous in their character to the tints of Newton’s scale. With his usual acuteness, Young at once attributed these colours to interference between the ordinary and extraordinary waves, and showed that the thickness of crystal required to develop a given tint, inversely proportional to the doubly refracting power, was in agreement with this view. But the complete explanation, demanding a fuller knowledge of the laws of interference of polarized light, was reserved for Fresnel and Arago. The subject is one which admits of great development; but the interest turns principally upon the beauty of the effects, and upon the facility with which many of them may be obtained in experiment. We must limit ourselves to a brief treatment of one or two of the simpler cases.

The incident vibration being plane-polarized, we will suppose that its plane makes an angle α with the principal plane of the crystal. On entering the crystal it is accordingly resolved into the two components represented by cos α cos φ, sin a cos φ, where φ = 2∙πt∣τ.

In traversing the crystal both waves are retarded, but we are concerned only with the difference of the retardations. Denoting the difference by p, we may take as the expressions of the waves on emergence cos a cos φ, sin a cos (φ - p).

It may be remarked that, in the absence of dispersion, p would be inversely proportional to λ; but in fact there are many cases where it deviates greatly from this law.

Now let the plane of analysation be inclined at the angle β to that of primitive polarization (fig. 25). Then for the sum of the two resolved components we have cos a cos (α - /3) cos φ + sin a sin (α - /3) cos {φ~ ρ), of which the intensity is

{cos a cos (a - β~) + sin a sin (a - β) cos ρ } 2 + sin 2 a sin 2 (a - β^) sin 2 β = cos 2 β-sin2asin2(a-j8)sin 2 ⅜p .... (1).

If in (1) we write β + ⅛π in place of β, we get sin 2 3 + sin2asin2(a-∣8)sin 2 ⅛p . . . . (2); and we notice that the sum of (1) and (2) is unity under all circumstances. The effect of rotating the analyser through 90° is thus always to transform the tint into its complementary. The two complementary tints may be seen at the same time if we employ a double image prism. In the absence of an analyser we may regard the two images as superposed, and there is no colour.

These expressions may be applied at once to the explanation of the colours of thin plates of mica or selenite. In this case the retardation p is proportional to the thickness, and approximately independent of the precise direction of the light, supposed to be nearly perpendicular to the plate, viz., nearly parallel to a principal axis of the crystal.

The most important cases are when β=0, 0=½π. In the latter the field would be dark were the plate removed; and the actual intensity is sin 2 2αsin 2½ρ (3).

The composition of the light is thus independent of the azimuth of the plate (a ); but the intensity varies greatly, vanishing four times during the complete revolution. The greatest brightness occurs when the principal plane bisects the angle between the planes of polarization and analysis. If β = 0, the light is complementary to that represented by (3).

If two plates be superposed, the retardations are added if the azimuths correspond; but they are subtracted if one plate be rotated relatively to the other through 90°. It is thus possible to obtain colour by the superposition of two nearly similar plates, although they may be too thick to answer the purpose separately.

If dispersion be neglected, the law of the colours in (3) is the same as that of the reflected tints of Newton’s scale. The thicknesses of the plates of mica (acting by double refraction) and of air required to give the same colour are as 400: 1. When a plate is too thick to show colour, its action may be analysed with the aid of a spectroscope.

Still thicker plates may be caused to exhibit colour, if the direction of the light within them makes but a small angle with an optic axis. Let us suppose that a plate of Iceland spar, or other uniaxal crystal (except quartz), cut perpendicularly to the axis, is interposed between the polarizing and analysing apparatus, and that the latter is so turned that the field is originally dark. The ray which passes perpendicularly is not doubly refracted, so that the centre of the field remains dark. At small angles to the optic axis the relative retardation is evidently proportional to the square oi the inclination, so that the colours are disposed in concentric rings. But the intensity is not the same at the various parts of the circumference. In the plane of polarization and in the perpendicular plane there is no double refraction, or rather one of the refracted rays vanishes. Along the corresponding lines in the field of view there is no revival of light, and the ring system is seen to be traversed by a black cross.

In many crystals the influence of dispersion is sufficient to sensibly modify the proportionality of p to λ. In one variety of uniaxal apophyllite Herschel found the rings nearly achromatic, indicating that p was almost independent of λ. Under these circumstances a much larger number of rings than usual became visible.

In biaxal crystals, cut so that the surfaces are equally inclined to the optic axes, the rings take the form of lemniscates.

A medium originally isotropic may acquire the doubly refracting property under the influence of strain; and, if the strain be homogeneous, the conditions are optically identical with those found in a natural crystal. The principal axes of the wave-surface coincide with those of strain. If the strain be symmetrical, the medium is optically uniaxal. In general, if P, Q, R be the principal stresses, the difference of velocities for waves propagated parallel to R is evidently proportional to (P - Q), and so on.

More often it happens that the strain is not homogeneous. Even then the small parts may be compared to crystals, but the optical constants vary from point to point. The comparatively feeble doubly refracting power thus developed in glass may best be made evident by the production of the colours of polarized light. Thus, in an experiment due to Brewster, a somewhat stout slab of glass, polished on the edges, is interposed between crossed Nicols. When the slab is bent in a plane perpendicular to that of vision, a revival of light takes place along the edges, where the elongation and contraction is greatest. If the width (in the direction of vision) be sufficient, the effect may be increased until the various colours of Newton’s scale are seen. These colours vary from point to point of the thickness in the plane of bending, the “neutral axis” remaining dark. The optic axis, being everywhere coincident with the direction of elongation (or contraction), is parallel to the length of the slab. To this direction the plane of polarization should be inclined at about 45°.

The condition of internal strain is not necessarily due to forces applied from without. Thus, if glass originally free from strain be unequally heated, the accompanying expansions give rise to internal strains which manifest themselves in polarized light. If the heating be moderate, so as not to approach the softening point, the state of ease is recovered upon cooling, and the double refraction disappears. But if the local temperature be raised further, the hot parts may relieve themselves of the temporary strain, and then upon cooling they and other parts may be left in a condition of permanent strain. Sudden cooling of glass heated to the softening point leads to a similar result. The outer parts harden while the interior is still at a higher temperature, so that, when the whole is cooled down, the outside, being as it were too large for the inside, is in a condition of radial tension and circumferential compression. An examination in polarized light shows that the strains thus occasioned are often very severe. If any small part be relieved by fracture from the constraint exercised upon it by the remainder, the doubly refracting property almost or wholly disappears. In this respect unannealed glass differs essentially from a crystal, all parts of which are similar and independent. It may be remarked that it is difficult to find large pieces of glass so free from internal strain as to show no revival of light when examined between crossed Nicols.

§ 23. Rotatory Polarization.

In general a polarized ray travelling along the axis of a uniaxal crystal undergoes no change; but it was observed by Arago that, if quartz be used in this experiment, the plane of polarization is found to be rotated through an angle proportional to the thickness of crystal traversed. The subject was further studied by Biot, who ascertained that the rotation due to a given thickness is inversely as the square of the wave-length of the light, thus varying very rapidly with the colour. In some, specimens of quartz (called in consequence right-handed) the rotation is to the right, while in others it is to the left. Equal thicknesses of right-and lefthanded quartz may thus compensate one another.

Fresnel has shown that the rotation of the plane may be interpreted as indicating a different velocity of propagation of the two circularly-polarized components into which plane-polarized light may always be resolved. In ordinary media the right-and lefthanded circularly-polarized rays travel at the same speed, and at any stage of their progress recompound a ray rectilinearly-polarized in a fixed direction. But it is otherwise if the velocities of propagation of the circular components be even slightly different.

The first circularly-polarized wave may be expressed by

ξι = rcos(ni-⅛ 1 z), 77 1 = r sin (nt - Zqz) . . . (1); and the second (of equal amplitude) by c 2 ≈r cοs(nt-k<f), <η 2 ≈ -rsin(nf -k 2 z) . . (2).

The resultant of (1) and (2) is t = ξι + <2 = 2i, cos⅛(⅛ 2 -⅛ 1 )zcos{ni-⅜(⅛ 1 + ⅛ 2 )z} , V = n 1 + ¾ = 2r sin ⅜(⅛ 2 - fc 1 ) z cos{ nt - l 1 (k γ + fc 1 )z}; so that

√ξ=tan⅛(⅛ 2 -⅛ 1 )z (3), which shows that for any fixed value of z the light is plane-polarized. The direction of this plane, however, varies with z. Thus, if η∕ξ= tan θ , so that θ gives the angular position of the plane in reference to ξ, we have θ = ½(k 2 - k 1 ) z (4), indicating a rotation proportional to z. The quantities k 1 , k 2 are inversely as the wave-lengths of the two circular components for the same periodic time. When the relative retardation amounts to an entire period, (k 2 - k 1 ) z= 2π, and then, by (4), θ = π. The revolution of the plane through two right angles restores the original state of polarization. In quartz the rotation is very rapid, amounting in the case of yellow light to about 24° for each millimetre traversed.

It is interesting to observe with what a high degree of accuracy the comparison of the velocities of the two waves can be effected. If the plane of polarization be determined to one minute of angle, a relative retardation of λ∕10800 is made manifest. If l be the thickness traversed, v and v + δv the two velocities, the relative retardation is lδν/υ. To take an example, suppose that l= 20 inches, λ = 1 / 40000 inch; so that if δ v∣v exceed 10 -8 , the fact might be detected.

In quartz the rotation of the plane depends upon the crystalline structure, but there are many liquids, e.g., oil of turpentine and common syrup, which exhibit a like effect. In such cases the rotation is of course independent of the direction of the light; it must be due to some peculiarity in the constitution of the molecules.

A remarkable connexion has been observed between the rotatory property and the crystalline form. Thus Herschel found that in many specimens the right-handed and left-handed varieties of quartz could be distinguished by the disposition of certain subordinate faces. The crystals of opposite kinds are symmetrical in a certain sense, but are yet not superposable. The difference is like that between otherwise similar right-and left-handed screws. The researches of Pasteur upon the rotatory properties of tartaric acid have opened up a new and most interesting field of chemistry. At that time two isomeric varieties were known,—ordinary tartaric acid, which rotates to the right, and racemic acid, which is optically inactive, properties of the acids shared also by the salts. Pasteur found that the crystals of tartaric acid and of the tartrates possessed a right-handed structure, and endeavoured to discover corresponding bodies with a left-handed structure. After many trials crystallizations of the double racemate of soda and ammonia were obtained, including crystals of opposite kinds. A selection of the right-handed specimens yielded ordinary dextro-tartaric acid, while a similar selection of the left-handed crystals gave a new variety— laevo-tartaric acid, rotating the plane of polarization to the left in the same degree as ordinary tartarie acid rotates it to the right. A mixture in equal proportions of the two kinds of tartaric acid, which differ scarcely at all in their chemical properties, reconstitutes racemic acid.

The possibility of inducing the rotatory property in bodies otherwise free from it was one of the finest of Faraday’s discoveries. He found that, if heavy glass, bisulphide of carbon, &c., are placed in a magnetic field, a ray of polarized light, propagated along the lines of magnetic force, suffers rotation. The laws of the phenomenon were carefully studied by Verdet, whose conclusions may be summed up by saying that in a given medium the rotation of the plane for a ray proceeding in any direction is proportional to the difference of magnetic potential at the initial and final points. In bisulphide of carbon, at 18° and for a difference of potential equal to unity C. G. S., the rotation of the plane of polarization of a ray of soda light is ⋅0402 minute of angle.

A very important distinction should be noted between the magnetic rotation and that natural to quartz, syrup, &c. In the latter the rotation is always right-handed or always left-handed with respect to the direction of the ray. Hence when the ray is reversed the absolute direction of rotation is reversed also. A ray which traverses a plate of quartz in one direction, and then after reflexion traverses the same thickness again in the opposite direction, recovers its original plane of polarization. It is quite otherwise with the rotation under magnetic force. In this case the rotation is in the same absolute direction even though the ray be reversed. Hence, if a ray be reflected backwards and forwards any number of times along a line of magnetic force, the rotations due to the several passages are all accumulated. The non-reversibility of light in a magnetized medium proves the case to be of a very exceptional character, and (as was argued by Thomson) indicates that the magnetized medium is itself in rotatory motion independently of the propagation of light through it.

The importance of polarimetric determinations has led to the contrivance of various forms of apparatus adapted to the special requirements of the case. If the light be bright enough, fairly accurate measurements may be made by merely rotating a Nicol until the field appears dark. Probably the best form of analyser, when white light is used and the plane is the same for all the coloured components, is the Jellet, formed by the combination of two portions of Iceland spar. By this instrument the field of view is duplicated, and the setting is effected by turning it until the two portions of the field, much reduced in brightness, appear equally dark. A similar result is attained in the Laurent, which, however, is only applicable to homogeneous light. In this apparatus, advantage is taken of the action of a half-wave plate. In passing such a plate the plane of polarization is as it were reflected by the principal section, that is, rotated until it makes the same angle with the principal section as at first, but upon the further side. The plate covers only half of the field of view, and the eye is focused upon the dividing edge. The planes of polarization of the two halves of the field are different, unless the original plane be parallel (or perpendicular) to the principal section. In the Laurent analyser the halfwave plate is rigidly combined with a Nicol in such a position that the principal section of the latter makes a small but finite angle with that of the plate. The consequence is that the two halves of the field of view cannot be blackened simultaneously, but are rendered equally dark when the instrument is so turned that the principal section of the plate is parallel to the plane of original polarization, which is also that of the uncovered half of the field. A slight rotation in either direction darkens one half of the field and brightens the other half.

In another form of “half-shade” polarimeter, invented by Poynting, the half-wave plate of the Laurent is dispensed with, a small rotation of one half of the field with respect to the other half being obtained by quartz (cut perpendicularly to the axis) or by syrup. In the simplest construction the syrup is contained in a small cell with parallel glass sides, and the division into two parts is effected by the insertion of a small piece of plate glass about 3 / 16 inch thick, a straight edge of which forms the dividing line. If the syrup be strong, the difference of thickness of 3 / 16 inch gives a relative rotation of about 2°. In this arrangement the sugar cell is a fixture, and only the Nicol rotates. The reading of the divided circle corresponds to the mean of the planes for the two halves of the field, and this of course differs from the original position of the plane before entering the sugar. This circumstance is usually of no importance, the object being to determine the rotation of the plane of polarization when some of the conditions are altered.

A discussion of the accuracy obtainable in polarimetry will be found in a recent paper by Lippich.

In Soleil's apparatus, designed for practical use in the estimation of the strength of sugar solutions, the rotation due to the sugar is compensated by a wedge of quartz. Two wedges, one of right-handed and the other of left-handed quartz, may be fitted together, so that a movement of the combination in either direction increases the thickness of one variety traversed and diminishes that of the other. The linear movement required to compensate the introduction of a tube of syrup measures the quantity of sugar present.

§ 24. Dynamical Theory of Diffraction.

The explanation of diffraction phenomena given by Fresnel and his followers is independent of special views as to the nature of the ether, at least in its main features; for in the absence of a more complete foundation it is impossible to treat rigorously the mode of action of a solid obstacle such as a screen. The full solution of problems of this kind is scarcely to be expected. Even in the much simpler case of sound, where we know what we have to deal with, the mathematical difficulties are formidable; and we are not able to solve even such an apparently elementary question as the transmission of sound past a rigid infinitely thin plane screen, bounded by a straight edge, or perforated with a circular aperture. But, without entering upon matters of this kind, we may inquire in what manner a primary wave may be resolved into elementary secondary waves, and in particular as to the law of intensity and polarization in a secondary wave as dependent upon its direction of propagation, and upon the character as regards polarization of the primary wave. This question is treated by Stokes in his “Dynamical Theory of Diffraction” on the basis of the elastic solid theory.

Let x, y, z be the coordinates of any particle of the medium in its natural state, and ξ, η, ζ the displacements of the same particle at the end of time t, measured in the directions of the three axes respectively. Then the first of the equations of motion may be put under the form

S =&2 (^ + B + S) +(a2_&2) i(S + S + i) ’ where α 2 and b 2 denote the two arbitrary constants. Put for shortness

⅛+⅛ + ‰δ (1), dx dy dz and represent by ▽ 2 ξ the quantity multiplied by b 2 . According to this notation, the three equations of motion are d 2 ξ 1Λ 2. t Z 2 12∖ rfδ Ί ⅛-S*v≈i + (α≈-¾ g- t ≈v⅛ + (α≡- δ >)g (2).

It is to be observed that δ denotes the dilatation of volume of the element situated at {x, y, z). In the limiting case in which the medium is regarded as absolutely incompressible δ vanishes; but, in order that equations (2) may preserve their generality, we must suppose a at the same time to become infinite, and replace α 2 δ by a new function of the coordinates.

These equations simplify very much in their application to plane waves. If the ray be parallel to OX, and the direction of vibration parallel to OZ, we have ξ = 0, η = 0, while ς is a function of x and t only. Equation (1) and the first pair of equations (2) are thus satisfied identically. The third equation gives f- ii 2 < 8 >-

of which the solution is

ς= f (bt - x ) (4)

where f is an arbitrary function.

The question as to the law of the secondary waves is thus answered by Stokes. “Let ξ = 0, η = 0, ζ=f(bt-x) be the displacements corresponding to the incident light; let O 1 be any point in the plane P (of the wave-front), d S an element of that plane adjacent to O 1 ; and consider the disturbance due to that portion only of the incident disturbance which passes continually across d S. Let O be any point in the medium situated at a distance from the point O 1 which is large in comparison with the length of a wave; let O 1 O = r , and let this line make an angle θ with the direction of propagation of the incident light, or the axis of x, and φ with the direction of vibration, or axis of z. Then the displacement at O will take place in a direction perpendicular to O 1 O, and lying in the plane ZO 1 O; and, if ζ' be the displacement at O, reckoned positive in the direction nearest to that in which the incident vibrations are reckoned positive,

C = 7^~-(l + cos0)sin<∕>∕ ’ (i>i-r) .

47TΓ

In particular, if

∕(δi-oe) = csin — (δi-∙τ) (5),

À

we shall have 2 ζ ’ = -fl + cοsθ')sinφcοs-(bt-r). . . (6).' ’

It is then verified that, after integration with respect to d S, (6) gives the same disturbance as if the primary wave had been supposed to pass on unbroken.

The occurrence of sin φ as a factor in (6) shows that the relative intensities of the primary light and of that diffracted in the direction θ depend upon the condition of the former as regards polarization. If the direction of primary vibration be perpendicular to the plane of diffraction (containing both primary and secondary rays), sin φ = 1; but, if the primary vibration be in the plane of diffraction, sin φ = cos θ. This result was employed by Stokes as a criterion of the direction of vibration; and his experiments, conducted with gratings, led him to the conclusion that the vibrations of polarized light are executed in a direction perpendicular to the plane of polarization.

The factor (1 + cos θ) shows in what manner the secondary disturbance depends upon the direction in which it is propagated with respect to the front of the primary wave.

If, as suffices for all practical purposes, we limit the application of the formulae to points in advance of the plane at which the wave is supposed to be broken up, we may use simpler methods of resolution than that above considered. It appears indeed that the purely mathematical question has no definite answer. In illustration of this the analogous problem for sound may be referred to. Imagine a flexible lamina to be introduced so as to coincide with the plane at which resolution is to be effected. The introduction of the lamina (supposed to be devoid of inertia) will make no difference to the propagation of plane parallel sonorous waves through the position which it occupies. At every point the motion of the lamina will be the same as would have occurred in its absence, the pressure of the waves impinging from behind being just what is required to generate the waves in front. Now it is evident that the aerial motion in front of the lamina is determined by what happens at the lamina without regard to the cause of the motion there existing. Whether the necessary forces are due to aerial pressures acting on the rear, or to forces directly impressed from without, is a matter of indifference. The conception of the lamina leads immediately to two schemes, according to which a primary wave may be supposed to be broken up. In the first of these the element d S the effect of which is to be estimated, is supposed to execute its actual motion, while every other element of the plane lamina is maintained at rest. The resulting aerial motion in front is readily calculated; it is symmetrical with respect to the origin, i.e., independent of θ. When the secondary disturbance thus obtaiued is integrated with respect to d S over the entire plane of the lamina, the result is necessarily the same as would have been obtained had the primary wave been supposed to pass on without resolution, for this is precisely the motion generated when every element of the lamina vibrates with a common motion, equal to that attributed to d S. The only assumption here involved is tho evidently legitimate one that, when two systems of variously distributed motion at the lamina are superposed, the corresponding motions in front are superposed also.

The method of resolution just described is the simplest, but it is only one of an indefinite number that might be proposed, and which are all equally legitimate, so long as the question is regarded as a merely mathematical one, without reference to the physical properties of actual screens. If, instead of supposing the motion at d S to be that of the primary wave, and to be zero elsewhere, we suppose the force operative over the element d S of the lamina to be that corresponding to the primary wave, and to vanish elsewhere, we obtain a secondary wave following quite a different law. In this case the motion in different directions varies as cos θ, vanishing at right angles to the direction of propagation of the primary wave. Here again, on integration over the entire lamina, the aggregate effect of the secondary waves is necessarily the same as that of the primary.

In order to apply these ideas to the investigation of the secondary wave of light, we require the solution of a problem, first treated by Stokes, viz., the determination of the motion in an infinitely extended elastic solid due to a locally applied periodic force. If we suppose that the force impressed upon the element of mass Ddx⅜dzis DZd⅛⅜<fc, being everywhere parallel to the axis of Z, the only change required in our equations (1), (2) is the addition of the term Z to the second member of the third equation (2). In the forced vibration, now under consideration, Z, and the quantities ξ, η, ς, δ expressing the resulting motion, are to be supposed proportional to e int , where i = √ (-1), and n =2π∕τ, τ being the periodic time. Under these circumstances the double differentiation with respect to t of any quantity is equivalent to multiplication by the factor - n 2 , and thus our equations take the form

(δ 2 V 2 + n 2 )ξ + (α 2 -δ 2 ) J=0

(δV + n≡), + (α 2 -J≡)g- 0 '. .... (7).

(δ 2 V a +n 2 )C+(α a -δ 2 )^≈-Z

It will now be convenient to introduce the quantities ῶ 1 , ῶ 2 , ῶ 3 , which express the rotations of the elements of the medium round axes parallel to those of coordinates, in accordance with the equations ⅛ - ⅛ „ dη dζ ~dζ dξ 3 dy dx , 1 dz dy ’ ^ a ' 2 dx dz * ’ -

In terms of these we obtain from (7), by differentiation and subtraction,

(δ 2 v 2 + n 2 )w 3 = 0 1

(δ 2 v 2 + n 2 )w 1 ·= dZ∣dy > (9).

(δ 2 v 2 + n 2 )≈r 2 = -dZ∕<ta )

The first of equations (9) gives

^3 ’ = 0 (lθ).

For τv 1 we have "'~^4⅛ffιff- dxdydz (11)· This solution may be verified in the same manner as Poisson’s theorem, in which k =0. " where r is the distance between the element dxdydz and the point where ῶ 1 is estimated, and

⅛=n∕δ = 2τr∕λ (12),

λ being the wave length.

We will now introduce the supposition that the force Z acts only within a small space of volume T, situated at (x, y, z), and for simplicity suppose that it is at the origin of coordinates that the rotations are to be estimated. Integrating by parts in (11), we get fe~* r d7 i , _C„c- ikr ~] fr, d (e- ikr ∖. J r dy c,j L z r J J 7, dy∖ r P ,j, in which the integrated terms at the limits vanish, Z being finite only within the region T. Thus

Since the dimensions of T are supposed to be very small in comparison with λ, the factor—- ∖ is sensibly constant; so that, if Z stand for the mean value of Z over the volume T, we may write ct ≈∑2 y d(e-<*r∖

4ιrδ 2 'r (13) ·

In like manner we find

TZ X die~ ikr ∖ , 1 ..

4πδ 2 r rfr∖ r )

From (10), (13), (14) we sec that, as might have been expected, the rotation at any point is about an axis perpendicular both to the direction of the force and to the line joining the point to the source of disturbance. If the resultant rotation be ῶ we have

_ TZ V(¾ a + y 2 ) d ∕e - ** r ∖ _ TZ sin φ d he^ ikr ∖ zr 4πδ 2 r dr∖ r J 4πδ 2 dr∖ r J ’ Φ denoting the angle between r and z. In differentiating e^ ttr ∕r with respect to r, we may neglect the term divided by r 2 as altogether insensible, kr being an exceedingly great quantity at any moderate distance from the origin of disturbance. Thus

— i⅛.TZsin≠ e^ ikr ^ a 4ÏÏP ~ (15) ’

which completely determines the rotation at any point. For a disturbing force of given integral magnitude it is seen to be everywhere about an axis perpendicular to r and the direction of the force, and in magnitude dependent only upon the angle (ϕ) between these two directions and upon the distance (r).

The intensity of light is, however, more usually expressed in terms of the actual displacement in the plane of the wave. This displacement, which we may denote by ζ', is in the plane containing z and r, and perpendicular to the latter. Its connexion with ῶ is expressed by ῶ =dζ , ∣dr; so that λ , TZsin≠ e <(n<-*r) χiftλ Ç - 4πδ 2 ' r t λ where the factor e int is restored.

Retaining only the real part of (16), we find, as the result of a local application of force equal to

DTZcos nt (17), the disturbance expressed by

., TZsin≠ cos(ni-⅛r) , ιc .

i 7 (18) ·

The occurrence of sinϕ shows that there is no disturbance radi ated in the direction of the force, a feature which might have been anticipated from considerations of symmetry.

We will now apply (18) to the investigation of a law of secondary disturbance, when a primary wave f= sin (nt - kx) (19)

is supposed to be broken up in passing the plane x =0. The first step is to calculate the force which represents the reaction between the parts of the medium separated by x =0. The force operative upon the positive half is parallel to OZ, and of amount per unit of area equal to - b-~Ddζ∣dx≈ δ 2 ΛD cos nt; and to this force acting over the whole of the plane the actual motion on the positive side may be conceived to be due. The secondary disturbance corresponding to the element d S of the plane may be supposed to be that caused by a force of the above magnitude acting over d S and vanishing elsewhere; and it only remains to examine what the result of such a force would be.

Now it is evident that the force in question, supposed to act upon the positive half only of the medium, produces just double of the effect that would be caused by the same force if the medium were undivided, and on the latter supposition (being also localized at a point) it comes under the head already considered. According to (18), the effect of the force acting at dS parallel to OZ, and of amount equal to

2δ 2 UDdS cos «ί, will be a disturbance

., dSsin<⅛ , , , ,

£ =—- cos tnt - kr) (20), regard being had to (12). This therefore expresses the secondary disturbance at a distance r and in a direction making an angle φ with OZ (the direction of primary vibration) due to the element d S of the wave-front.

The proportionality of the secondary disturbance to sinϕ is common to the present law and to that given by Stokes, but here there is no dependence upon the angle θ between the primary and secondary rays. The occurrence of the factor (λr) -1 , and the necessity of supposing the phase of the secondary wave accelerated by a quarter of an undulation, were first established by Archibald Smith, as the result of a comparison between the primary wave, supposed to pass on without resolution, and the integrated effect of all the secondary waves (§ 10). The occurrence of factors such as sinϕ, or ½(1+cos θ ), in the expression of the secondary wave has no influence upon the result of the integration, the effects of all the elements for which the factors differ appreciably from unity being destroyed by mutual interference.

The choice between various methods of resolution, all mathematically admissible, would be guided by physical considerations respecting the mode of action of obstacles. Thus, to refer again to the acoustical analogue in which plane waves are incident upon a perforated rigid screen, the circumstances of the case are best represented by the first method of resolution, leading to symmetrical secondary waves, in which the normal motion is supposed to be zero over the unperforated parts. Indeed, if the aperture is very small, this method gives the correct result, save as to a constant factor. In like manner our present law (20) would apply to the kind of obstruction that would be caused by an actual physical division of the elastic medium, extending over the whole of the area supposed to be occupied by the intercepting screen, but of course not extending to the parts supposed to be perforated. In the present state of our ignorance this law seems to be at least as plausible as any other.

§ 25. The Diffraction of Light by Small Particles.

The theory of the diffraction, dispersion, or scattering of light by small particles, as it has variously been called, is of importance, not only from its bearings upon fundamental optical hypotheses, but on account of its application to explain the origin and nature of the light from the sky. The view, suggested by Newton and advocated in more recent times by such authorities as Herschel and Clausius, that the light of the sky is a blue of the first order reflected from aqueous particles, was connected with the then prevalent notion that the suspended moisture of clouds and mists was in the form of vesicles or bubbles. Experiments such as those of Brücke pointed to a different conclusion. When a weak alcoholic solution of mastic is agitated with water, the precipitated gum scatters a blue light, obviously similar in character to that from the sky. Not only would it be unreasonable to attribute a vesicular structure to the mastic, but (as Brücke remarked) the dispersed light is much richer in quality than the blue of the first order. Another point of great importance is well brought out in the experiments of Tyndall, vol. cxxxvii. p. 388. ] upon clouds precipitated by the chemical action of light. Whenever the particles are sufficiently fine, the light emitted laterally is blue in colour, and, in a direction perpendicular to the incident beam, is completely polarized

About the colour there can be no prima facie difficulty; for, as soon as the question is raised, it is seen that the standard of linear dimension, with reference to which the particles are called small, is the wave-length of light, and that a given set of particles would (on any conceivable view as to their mode of action) produce a continually increasing disturbance as we pass along the spectrum towards the more refrangible end.

On the other hand, that the direction of complete polarization should be independent of the refracting power of the matter composing the cloud has been considered mysterious. Of course, on the theory of thin plates, this direction would be determined by Brewster’s law; but, if the particles of foreign matter are small in all their dimensions, the circumstances are materially different from those under which Brewster’s law is applicable.

The investigation of this question upon the elastic solid theory will depend upon how we suppose the solid to vary from one optical medium to another. The slower propagation of light in glass or water than in air or vacuum may be attributed to a greater density, or to a less rigidity, in the former case; or we may adopt the more complicated supposition that both these quantities vary, subject only to the condition which restricts the ratio of velocities to equality with the known refractive index. It will presently appear that the original hypothesis of Fresnel, that the rigidity remains the same in both media, is the only one that can be reconciled with the facts; and we will therefore investigate upon this basis the nature of the secondary waves dispersed by small particles.

Conceive a beam of plane polarized light to move among a number of particles, all small compared with any of the wavelengths. According to our hypothesis, the foreign matter may be supposed to load the aether, so as to increase its inertia without altering its resistance to distortion. If the particles were away, the wave would pass on unbroken and no light would be emitted laterally. Even with the particles retarding the motion of the aether, the same will be true if, to counterbalance the increased inertia, suitable forces are caused to act on the aether at all points where the inertia is altered. These forces have the same period and direction as the undisturbed luminous vibrations themselves. The light actually emitted laterally is thus the same as would be caused by forces exactly the opposite of these acting on the medium otherwise free from disturbance, and it only remains to see what the effect of such force would be.

On account of the smallness of the particles, the forces acting throughout the volume of any individual particle are all of the same intensity and direction, and may be considered as a whole. The determination of the motion in the aether, due to the action of a periodic force at a given point, is a problem with which we have recently been occupied (§ 24). But, before applying the solution to a mathematical investigation of the present question, it may be well to consider the matter for a few moments from a more general point of view.

In the first place, there is necessarily a complete symmetry round the direction of the force. The disturbance, consisting of trans verse vibrations, is propagated outwards in all directions from the centre; and, in consequence of the symmetry, the direction of vibration in any ray lies in the plane containing the ray and the axis of symmetry; that is to say, the direction of vibration in the scattered or diffracted ray makes with the direction of vibration in the incident or primary ray the least possible angle. The symmetry also requires that the intensity of the scattered light should vanish for the ray which would be propagated along the axis; for there is nothing to distinguish one direction transverse to the ray from another. The application of this is obvious. Suppose, for distinctness of statement, that the primary ray is vertical, and that the plane of vibration is that of the meridian. The intensity of the light scattered by a small particle is constant, and a maximum, for rays which lie in the vertical plane running east and west, while there is no scattered ray along the north and smith line. If the primary ray is unpolarized, the light scattered north and south is entirely due to that component which vibrates east and west, and is therefore perfectly polarized, the direction of its vibration being also east and west. Similarly any other ray scattered horizontally is perfectly polarized, and the vibration is performed in the horizontal plane. In other directions the polarization becomes less and less complete as we approach the vertical.

The observed facts as to polarization are thus readily explained, and the general law connecting the intensity of the scattered light with the wave-length follows almost as easily from considerations of dimensions.

The object is to compare the intensities of the incident and scattered light, for these will clearly be proportional. The number (i ) expressing the ratio of the two amplitudes is a function of the following quantities :—(T) the volume of the disturbing particle; (r) the distance of the point under consideration from it; (λ) the wave-length; (b ) the velocity of propagation of light; (D) and (D') the original and altered densities: of which the first three depend only upon space, the fourth on space and time, while the fifth and sixth introduce the consideration of mass. Other elements of the problem there are none, except mere numbers and angles, which do not depend upon the fundamental measurements of space, time, and mass. Since the ratio (i ), whose expression we seek, is of no dimensions in mass, it follows at once that D and D' occur only under the form D: D', which is a simple number and may therefore be disregarded. It remains to find how i varies with T, r, λ, b.

Now, of these quantities, b is the only one depending on time; and therefore, as i is of no dimensions in time, b cannot occur in its expression.

Moreover, since the same amount of energy is propagated across all spheres concentric with the particle, we recognize that i varies as r. It is equally evident that i varies as T, and therefore that it must be proportional to T∕λ r , T being of three dimensions in space. In passing from one part of the spectrum to another, λ is the only quantity which varies, and we have the important law :—

When light is scattered by particles which are very small compared with any of the wave-lengths, the ratio of the amplitudes of the vibrations of the scattered and incident lights varies inversely as the square of the wave-length, and the ratio of intensities as the inverse fourth power.

The light scattered from small particles is of a much richer blue than the blue of the first order as reflected from a very thin plate. From the general theory (§ 8), or by the method of dimensions, it is easy to prove that in the latter case the intensity varies as λ -2 , instead of λ -4 .

The principle of energy makes it clear that the light emitted laterally is not a new creation, but only diverted from the main stream. If I represent the intensity of the primary light after traversing a thickness x of the turbid medium, we have d I = -h I λ -4 dx, where h is a constant independent of λ. On integration, log(I∕I 0 )= -hλ -4 x (1), if I 0 correspond to x =0,—a law altogether similar to that of absorption, and showing how the light tends to become yellow and finally red as the thickness of the medium increases.

Captain Abney has found that the above law agrees remarkably well with his observations on the transmission of light through water in which particles of mastic are suspended.

We may now investigate the mathematical expression for the disturbance propagated in any direction from a small particle upon which a beam of light strikes. Let the particle be at the origin of coordinates, and let the expression for the primary vibration be ς=sin(nt-kx ) (2).

The acceleration of the element at the origin is - n 2 sin nt; so that the force which would have to be applied to the parts where the density is D' (instead of D), in order that the waves might pass on undisturbed, is per unit of volume,

- (D' - D) n 2 sin nt .

To obtain the total force which must be supposed to act, the factor T (representing the volume of the particle) must be introduced. The opposite of this, conceived to act at O, would give the same disturbance as is actually caused by the presence of the particle. Thus by (18) (§ 24) the secondary disturbance is expressed by λ , D' - D n 2 T sin φ sin (nt - kr) ζ = ~D Urδ 2 r

^D2-DπTsinφ sin ^, z . r ) (3). `

D λ 2 r

The preceding investigation is based upon the assumption that in passing from one medium to another the rigidity of the aether does not change. If we forego this assumption, the question is necessarily more complicated; but, on the supposition that the changes of rigidity (∆N) and of density (∆I)) are relatively small, the results are fairly simple. If the primary wave be represented by ζ=e -ikx (4), the component rotations in the secondary wave are

„ ∕∆D y ∆N xy∖ ...

^=‰7÷N√λ · · · · (5), T . ∕ ∆D X ∆N s 2 -≈ 2 ∖ wa ° p ∖"^D^ r + ΙΓ ~P~ J . where p i⅛ 3 T ∕ 6 x

4ιr r

The expression for the resultant rotation in the general case would be rather complicated, and is not needed for our purpose. It is easily seen to be about an axis perpendicular to the scattered ray (x, y, z), inasmuch as x ῶ 1 + y ῶ 2 + z ῶ 3 =0.

Let us consider the more special case of a ray scattered normally to the incident ray, so that x=0. We have

√∙-≈, 1 ≈+⅛ + ⅛-Γ=(^)⅛ + P≈(^)⅛ s . . (7).

If ∆N, ∆D be both finite, we learn from (7) that there is no direction perpendicular to the primary (polarized) ray in which the secondary light vanishes. Now experiment tells us plainly that there is such a direction, and therefore we are driven to the conclusion that either ∆N or ∆D must vanish.

The consequences of supposing ∆N to be zero have already been traced. They agree very well with experiment, and require us to suppose that the vibrations are perpendicular to the plane of polarization. So far as (7) is concerned the alternative supposition that ∆D vanishes would answer equally well, if we suppose the vibrations to be executed in the plane of polarization; but let us now revert to (5), which gives

P∆N yz P∆N xy P∆N z 2 -z 2 wa ~ - TΓr≡' w2 ~ + "n"^7*~ · (8) ·

According to these equations there would be, in all, six directions from O along which there is no scattered light,—two along the axis of y normal to the original ray, and four (y = 0, z = ± x ) at angles of 45° with that ray. So long as the particles are small no such vanishing of light in oblique directions is observed, and we are thus led to the conclusion that the hypothesis of a finite ∆N and of vibrations in the plane of polarization cannot be reconciled with the facts. No form of the elastic solid theory is admissible except that in which the vibrations are supposed to be perpendicular to the plane of polarization, and the difference between one medium and another to be a difference of density only. Before leaving this subject it may be instructive to show the application of a method, similar to that used for small particles, to the case of an obstructing cylinder, whose axis is parallel to the fronts of the primary waves. We will suppose (1) that the variation of optical properties depends upon a difference of density (D' —D), and is small in amount; (2) that the diameter of the cylinder is very small in comparison with the wave-length of light.

Let the axis of the cylinder be the axis of z (fig. 26), and (as before) let the incident light be parallel to x. The original vibration is thus, in the principal cases, parallel to either z or y. We will take first the former case, where the disturbance due to the cylinder must evidently be symmetrical round OZ and parallel to it. The element of the disturbance at A, due to PQ (dz), will be proportional to dz in amplitude, and will be retarded in phase by an amount corresponding to the distance r. In calculating the effect of the whole bar we have to consider the integral ∕^ co dz . . 7 . ∕* 00 <⅛sin (nt-kr)

∕ — sin (nt - kr) = ∕ 7τ ÷— v ., — .

√0 τ √r √(r 2 -K-)

The integral on the left may be treated as in § 15, and we find

Zoo r ~ 1 sin (nt - kr) dz≈∖J(κ(∖V) ’ SΛ.'a. (nt - ⅛R - ⅛7r), oe showing that the total effect is retarded ⅛λ behind that due to the central element at O. We have seen (3) that, if σ be the sectional area, the effect of the element PQ is

D'- D 7r<riZzsin <f> . , , , .

—p — sιn(m!-⅛∙), where φ is the angle OPA. In strictness this should be reckoned perpendicular to PA, and therefore, considered as a contribution to the resultant at A, should be multiplied by sin φ. But the factor sin 2 ϕ , being sensibly equal to unity for the only parts which are really operative, may be omitted without influencing the result. In this way we find, for the disturbance at A,

P⅛θ ™ sin(ni-≡-iπ) .... (9), corresponding to the incident wave sin (nt - kx).

When the original vibration is parallel to y, the disturbance due to the cylinder will no longer be symmetrical about OZ. If α be the angle between OX and the scattered ray, which is of course always perpendicular to OZ, it is only necessary to introduce the factor cos α in order to make the previous expression (9) applicable.

The investigation shows that the light diffracted by an ideal wiregrating would, according to the principles of Fresnel, follow the law of polarization enunciated by Stokes. On the other hand, this law would be departed from were we to suppose that there is any difference of rigidity between the cylinder and the surrounding medium.

§ 26. Reflexion and Refraction.

So far as the directions of the rays are concerned, the laws of reflexion and refraction were satisfactorily explained by Huygens on the principles of the wave theory. The question of the amount of light reflected, as dependent upon the characters of the media and upon the angle of incidence, is a much more difficult one, and cannot be dealt with a priori without special hypotheses as to the nature of the luminous vibrations, and as to the cause of the difference between various media. By a train of reasoning, not strictly dynamical, but of great ingenuity, Fresnel was led to certain formulae, since known by his name, expressing the ratio of the reflected to the incident vibration in terms of one constant (μ). If θ be the angle of incidence and θ 1 the angle of refraction, Fresnel’s expression for light polarized in the plane of incidence is sin(0-0 1 ) /n sin(0 + 0 1 ) v h where the relation between the angles θ, θ 1 , and μ (the relative refractive index) is, as usual, sin0 = μsin0 1 (2).

In like manner, for light polarized perpendicularly to the plane of incidence, Fresnel found tan(0-0 1 ) (3) tan(0 + 0 1 )

In the particular case of perpendicular incidence, both formulae coincide with one previously given by Young, viz.,

(μ-l)∕(μ + l). . . . . . . (4).

Since these formulae agree fairly well with observation, and are at any rate the simplest that can at all represent the facts, it may be advisable to consider their significance a little in detail. As θincreases from 0 to ½ π, the sine-formula increases from Young’s value to unity. We may see this most easily with the aid of a slight transformation :—

sin(0-0 1 ) 1 - tan 0 1 ∕ tan 0 μ - cos 0/ cos 0 j sin(0 + 0 1 ) l+tan0 1 ∕tan0 μ + cos 0/cos 0 1 ’ Now, writing cos0∕cos0 1 in the form

∕ ( 1 - sin 2 θ ) ∖∕ I 1 - μ^ 2 sin 2 0 ) ’ we recognize that, as θ increases from 0 to ½ π, cosθ∕cosθ 1 diminishes continuously from 1 to 0, and therefore (1) increases from (μ-l)∕(μ + l) to unity.

It is quite otherwise with the tangent-formula. Commencing at Young’s value, it diminishes, as θ increases, until it attains zero, when θ + θ 1 =½π, or sinθ 1 = cosθ; or by (2) tanθ = μ. This is the polarizing angle defined by Brewster. It presents itself here as the angle of incidence for which there is no reflexion of the polarized light under consideration. As the angle of incidence passes through the polarizing angle, the reflected vibration changes sign, and increases in numerical value until it attains unity at a grazing incidence (θ = ½π).

We have hitherto supposed that the second medium (into which the light enters at the refracting surface) is the denser. In the contrary case, total reflexion sets in as soon as sin θ =μ -1 , at which point θ 1 becomes imaginary. We shall be able to follow this better in connexion with a mechanical theory.

If light falls upon the first surface of a parallel plate at the polarizing angle, the refracted ray also meets the second surface of the plate at the appropriate polarizing angle. For if μ be the index of the second medium relatively to the first, the tangent of the angle of incidence, which is also the cotangent of the angle of refraction, is equal to μ. At the second surface (the third medium being the same as the first) the angles of incidence and refraction are interchanged, aud therefore the condition for the polarizing angle is satisfied, since the index for the second refraction is μ -1 .

The principal formulae apply to light polarized in, and perpendicular to, the plane of incidence. If the plane of polarization make an angle a with that of incidence, the original vibration may be resolved into two,—cos a polarized in the plane of incidence, and sin a polarized in the perpendicular plane. These components are reflected according to the laws already considered, and reconstitute plane-polarized light, of intensity cos ≈ o ⅛4^Z⅛) + sw t⅛g→l) . . .' (5).

srn 2 (0 + 0 1 ) tan 2 (0 + 0 1 )

If the incident light be polarized in a plane making 45° with the plane of incidence, or be circularly-polarized (§ 20), or be unpolarized, (5) applies to the reflected light, with substitution of ½ for cos 2 a and sin 2 α. If β denote in the general case the angle between the plane of incidence and that in which the reflected light is polarized,

, , cos (0 + 0.)

tan β = tan a — κ -—- l ( (6), cos(0-0 1 ) v ,, a result the approximate truth of which has been verified by Fresnel and Brewster.

The formulae for the intensities of the refracted light follow immediately from the corresponding formulae relative to the reflected light in virtue of the principle of energy. The simplest way to regard the matter is to suppose the refracted light to emerge from the second medium into a third medium similar to the first without undergoing loss from a second reflexion, a supposition which would be realized if the transition between the two media were very gradual instead of abrupt. The intensities of the different lights may then be measured in the same way; and the supposition that no loss of energy is incurred when the incident light gives rise to the reflected and refracted lights requires that the sum of the squares of the vibrations representing the latter shall be equal to the square of the vibration representing the former, viz., unity. We thus obtain, in the two cases corresponding to (1) and (3),

_ _ sin 2 (0-0 1 ) _ sin 20 sin 20 1 sin 2 (0 + 0 1 ) sin 2 (0 + 0 1 ) ' ' ,

. _ tan 2 (0 - 0 1 ) sin 20 sin 20 1 tan 2 (0 + 0 1 ) sin 2 (0 + 0 1 )cos 2 (0-0 1 ) ∙ ’ ∙('∙

A plate of glass, or a pile of parallel plates, is often convenient as a polarizer, when it is not necessary that the polarization be quite complete. At the precise angle of incidence (tan^ 1 μ) there would be, according to Fresnel’s formulae, only one kind of polarized light reflected, even when the incident light is unpolarized. The polarization of the transmitted light, on the other hand, is imperfect; but it improves as the number of plates is increased.

If we suppose that there is no regular interference, the intensity (r) of the light reflected from a plate is readily calculated by a geometric series when the intensity (ρ) of the light reflected from a single surface is known. The light reflected from the first surface is p. That transmitted by the first surface, reflected at the second, and then transmitted at the first, is ρ(1 - ρ) 2 . The next component, reflected three times and transmitted twice, is ρ 3 (1 - ρ) 2 , and so on. Hence r=p + (l-p) 2 { p + p8 4 y + ... }--½L . . . (9) .

The intensity of the light reflected from a pile of plates has been investigated by Provostaye and Desains. If φ(m) be the reflexion from m plates, we may find as above for the reflexion from (m + 1) plates,

≠(wι + l) = r+(l -r) 2 √>(τn){l +r≠(wι) + r 2 [<∕>(ιn)] 2 + J∙ 3 + .. .} r + (l - 2r)≠(τ⅞)

1 - rφ(m)

By means of this expression we may obtain in succession the values of ϕ (2), ϕ (3), &c., in terms of ϕ (1), viz., r. The general value is ≠< m >-∏√≡⅛- < 10 >∙

as may easily be verified by substitution.

The corresponding expression for the light transmitted by a pile of m plates is

⅜w-ww- 1+t 1 m ∖ r υ ,. .... di;.

The investigation has been extended by Stokes so as to cover the case in which the plates exercise an absorbing influence.

The verification of Fresnel’s formulae by direct photometric measurement is a matter of some difficulty. The proportion of perpendicularly incident light transmitted by a glass plate has been investigated by Rood; but the deficiency may have been partly due to absorption. If we attempt to deal directly with the reflected light, the experimental difficulties are much increased; but the evidence is in favour of the approximate correctness of Fresnel’s formulae when light is reflected nearly perpendicularly from a recently polished glass surface. When the surface is old, even though carefully cleaned, there may be a considerable falling off of reflecting power.

We have seen that according to Fresnel’s tangent-formula there would be absolutely no reflexion of light polarized perpendicularly to the plane of incidence, when the angle of incidence is tan -1 μ, or, which comes to the same thing, common light reflected at this angle could be perfectly extinguished with a Nicol’s prism.

It was first observed by Airy that in the case of the diamond and other highly refracting media this law is only approximately in accordance with the facts. It is readily proved by experiment that, whatever be the angle of incidence, sunlight reflected from a plate of black glass is incapable of being quenched by a Nicol, and is therefore imperfectly plane-polarized.

This subject has been studied by Jamin. The character of the reflected vibration can be represented, as regards both amplitude and phase, by the situation in a plane of a point P relatively to the origin of coordinates O. The length of the line OP represents the amplitude, while the inclination of OP to the axis of x represents the phase. According to Fresnel’s formula appropriate to light polarized perpendicularly to the plane of incidence, P is situated throughout on the axis of x, passing through O when the angle of incidence is tan -1 μ. Jamin found, however, that in general P does not pass through O, but above or below it. When P is on the axis of y, the amplitude is a minimum, and the phase is midway between the extreme phases. For one class of bodies the phase is in arrear of that corresponding to perpendicular incidence, and for another class of bodies in advance. In a few intermediate cases P passes sensibly through O; and then the change of phase is sudden, and the minimum amplitude is zero.

In the case of metals the polarization produced by reflexion is still more incomplete. Light polarized perpendicularly to the plane of incidence is reflected at all angles, the amount, however, decreasing as the angle of incidence increases from 0° to about 75°, and then again increasing up to a grazing incidence. The most marked effect is the relative retardation of one polarized component with respect to the other. At an angle of about 75° this retardation amounts to a quarter period.

The intensity of reflexion from metals is often very high. From silver, even at perpendicular incidence, as much as 95 per cent. of the incident light is reflected. There is reason for regarding the high reflecting power of metals as connected with the intense absorption which they exercise. Many aniline dyes reflect in abnormal proportion from their surface those rays of the spectrum to which they are most opaque. The peculiar absorption spectrum of permanganate of potash is reproduced in the light reflected from a surface of a crystal.

§ 27. Reflexion on the Elastic Solid Theory.

On the theory which assimilates the aether to an elastic solid, the investigation of reflexion and refraction presents no very serious difficulties, but the results do not harmonize very well with optical observation. It is, however, of some importance to understand that reflexion and refraction can be explained, at least in their principal features, on a perfectly definite and intelligible theory, which, if not strictly applicable to the aether, has at any rate a distinct mechanical significance. The refracting surface and the wave-fronts may for this purpose be supposed to be plane.

When the vibrations are perpendicular to the plane of incidence (z =0), the solution of the problem is very simple. We suppose that the refracting surface is x=0, the rigidity and density in the first medium being N, D, and in the second N 1 , D 1 . The displace ments in the two media are in general denoted by ξ,η,ζ; ξ υ η 1 ,; but in the present case ξ, η, ξ 1 , η 1 all vanish. Moreover ζ, Ϛ 1 are independent of z. The equations to be satisfied in the interior of the media are accordingly (§ 24)

Ö ≈⅛(^÷≤0 (I), dk U∖dxdy-J <P⅛ N 1 WιΛΛ , 2) dk V)'∖dx i df ) κ ''

At the boundary the conditions to be satisfied are the continuity of displacement and of stress; so that, when x =0, f-δ. n⅛-n,½ (S).

The incident waves may be represented by ζ= e Haχ+by+ct), where

Dc 2 =N(α 2 + δ 2 ) (4); and ax+by = const, gives the equation of the wave-fronts. The reflected aud refracted waves may be represented by

^∙=^ ’ gi(-a≈+ 1, i'+c<) (5),

(6).

The coefficient of t is necessarily the same in all three waves on account of the periodicity, and the coefficient of y must be the same since the traces of all the waves upon the plane of separation must move together. With regard to the coefficient of x, it appears by substitution in the differential equations that its sign is changed in passing from the incident to the reflected wave; in fact c 2 =V 2 {(±α) 2 + δ 2 } =V 1 2 {α 1 2 + δ 2 } .... (7), where V, V 1 are the velocities of propagation in the two media given by

V 2 =N∕D, V 1 2 = N 1 ∕D 1 (8).

Now δ∕χ∕(α 2 + δ 2 ) is the sine of the angle included between the axis of x and the normal to the plane of waves—in optical language, the sine of the angle of incidence—and δ∕√(α 1 2 + δ 2 ) is in like manner the sine of the angle of refraction. If these angles be denoted (as before) by θ, θ 1 , (7) asserts that sinØ: sin0 i is equal to the constant ratio V: V 1 , the well-known law of sines. The laws of reflexion and refraction follow simply from the fact that the velocity of propagation normal to the wave-fronts is constant in each medium, that is to say, independent of the direction of the wave-front, taken in connexion with the equal velocities of the traces of all the waves on the plane of separation (V∕sin0 = V 1 ∕sin0 1 ).

The boundary conditions (3) now give l + r=G', Nα(l-C)=N 1 α 1 <∕ · · · (9).

whence z , _ Nα — N 1 α 1 i Nα + N 1 α 1 U ' ’ a formula giving the reflected wave in terms of the incident wave (supposed to be unity). This completes the symbolical solution. If α 1 (and 0 1 ) be real, we see that, if the incident wave be

£≈ cos (ax + by + et), or in terms of V, λ, and 0,

2ιr

£=cos — (rr cos 0 + y sin 0 + V7) . . . . (ll),

À

the reflected wave is

. Ncot0-N 1 cot0 1 2τr. . , . a , .... ,. n .

f=-vr—τ~—xr^—rwcos — ( -a:cosØ + ysinØ+ ∖t) . (12). i, N cot 0 + N 1 cot 0 1 λ j k

The formula for intensity of the reflected wave is here obtained on the supposition that the waves are of harmonic type; but, since it does not involve λ and there is no change of phase, it may be extended by Fourier’s theorem to waves of any type whatever. It may be remarked that when the first and second media are interchanged the coefficient in (12) simply changes sign, retaining its numerical value.

The amplitude of the reflected wave, given in general by (12), assumes special forms when we introduce more particular suppositions as to the nature of the difference between media of diverse refracting power. According to Fresnel and Green the rigidity does not vary, or N = N 1 . In this case

N cot 0 - N 1 cot θ 1 _ cot 0 - cot 0 1 sin (0 1 - 0) Ncot0 + N 1 cot0 cot0 + cot0 1 sin(0 i + 0)'

If, on the other hand, the density is the same in various media,

N 1 : N = V 1 2 : V 2 — sin 2 0 1 : sin 2 0, and then N cot 0 - N 1 cot 0 1 tan(0 1 -0) N cot 0 + N 1 cot 0 1 tan (0 1 + 0) ’

If we assume the complete accuracy of Fresnel’s expressions, either alternative agrees with observation: only, if N = N 1 , light must be supposed to vibrate normally to the plane of polarization; while, if D = D υ the vibrations are parallel to that plane.

An intermediate supposition, according to which the refraction-is regarded as due partly to a difference of density and partly to a difference of rigidity, could scarcely be reconciled with observation, unless one variation were very subordinate to the other. But the most satisfactory argument against the joint variation is that derived from the theory of the diffraction of light by small particles (§ 25).

We will now, limiting ourselves for simplicity to Fresnel’s supposition (N 1 = N), inquire into the character of the solution when total reflexion sets in. The symbolical expressions for the reflected and refracted waves are ζ =⅛^ eii ' ax+bl ' +ct ' (13).

c f (-1

< 1 = e i(α 1 x+6y+ct) (14), 1 a+a 1 x ' and so long as a 1 is real they may be intrepreted to indicate f=~j~ cos (~ αz +^+ ci ) (15 )> ζι≈^⅛cοs(a 1 x+by+ct) (16), corresponding to the incident wave f=cos (ccx + by+ct) (17).

In this case there is a refracted wave of the ordinary kind, conveying away a part of the original energy. When, however, the second medium is the rarer (V 1 >V), and the angle of incidence exceeds the so-called critical angle [sin^ 1 (V∕V 1 )], there can be no refracted wave of the ordinary kind. In whatever direction it may be supposed to lie, its trace must necessarily outrun the trace of the incident wave upon the separating surface. The quantity α 1 , as defined by our equations, is then imaginary, so that (13) and (14) no longer express the real parts of the symbolical expressions (5) and (6).

If-τα 1 ' be written in place of a 1 , the symbolical equations are λO+⅛, <(- ox+ ⅛+rt) Λ ⅛L- e i(. ia l 'x+by+ct) . b aia 1 ' a-ia{ from which, by discarding the imaginary parts, we obtain C=∙cos(-o∞ + δy + ci + 2e) (18), ^~√(JL,v l, ' , ∙ c ° smcl+ ' 1 ∙ · · (1S> ’ where tan ∈ = α 1 ’ ∕α (20).

Since x is supposed to be negative in the second medium, we see that the disturbance is there confined to a small distance (a few wave-lengths) from the surface, and no energy is propagated into the interior. The whole of the energy of the incident waves is to be found in the reflected waves, or the reflexion is total. There is, however, a change of phase of 2e, given by (20), or in terms of V, V 1 , and 0

tane = √{tan 2 0-sec 2 0(V 2 ∕V 1 2 )} .... (21).

The principal application of the formulae being to reflexions when the second medium is air, it will be convenient to denote by μ the index of the first medium relatively to the second, so that μ=V 1 ∕V. Thus tan∈ = √{tan 2 0-sec 2 0∕ j u 2 } .... (22).

The above interpretation of his formula sin (0 1 - 0)∕sin (0 1 + 0), in the case where 0 1 becomes imaginary, is due to the sagacity of Fresnel. His argument was perhaps not set forth with full rigour, but of its substantial validity there can be no question. By a similar process Fresnel deduced from his tangent-formula for the change of phase (2√) accompanying total reflexion when the vibrations are executed in the plane of incidence, tan e ’ =μ√{μ 2 tan 2 0-sec 2 0} .... (23).

The phase-differences represented by 2e and 2∈' cannot be investigated experimentally, but the difference (2√ - 2e) is rendered evident when the incident light is polarized obliquely so as to contribute components in both the principal planes. If in the act of reflexion one component is retarded more or less than the other, the resultant light is no longer plane but elliptically polarized. .

From (22) and (23) we have tan(e'-e) = cos0√{l-μ^ 2 cθ3ec 2 0} , whence

.., n , o χ 2 m 2 sin 4 0-(l+ j u 2 ) sin 2 0 + 1 λ cos(2e-2e) (l·+μ 2) sin V∏ * « · (24 >∙

The most interesting case occurs when the difference of phase amounts to a quarter of a period, corresponding to light circularly polarized. If, however, we put cos (2e' - 2«) = 0, we find 4μ 2 sin 2 0 = l +μ 2 ±√{(l + μ 2 ) 2 ~ ⅜ 2 } »

from which it appears that, in order that sin 0 may be real, μ 2 must exceed 3 + √8. So large a value of μ 2 not being available, the conversion of plane-polarized into circularly-polarized light by one reflexion is impracticable.

The desired object may, however, be attained by two successive reflexions. The angle of incidence may be so accommodated to the index that the alteration of phase amounts to ⅛ period, in which case a second reflexion under the same conditions will give rise to light circularly polarized. Putting (2∈- 2ϵ') = ¼π we get

2 j u 2 sin 4 0 = (l + √⅛){(l+μ 2 )sin 2 0 + l} . . . (25), an equation by which θ is determined when μ is given.

It appears that, when ∕x = l"51, 0 = 48° 37' or 54° 37'.

These results were verified by Fresnel by means of the rhomb shown in fig. 27.

The problem of reflexion upon the elastic solid theory, when the vibrations are executed in the plane of incidence, is more complicated, on account of the tendency to form waves of dilatation. In order to get rid of these, to which no optical phenomena correspond, it is necessary to follow Green in supposing that the velocity of such waves is infinite, or that the media are incompressible. Even then we have to introduce in the neighbourhood of the interface waves variously called longitudinal, pressurai, or surface waves; otherwise it is impossible to satisfy the conditions of continuity of strain and stress. These waves, analogous in this respect to those occurring in the second medium when total reflexion is in progress (19), extend to a depth of a few wave-lengths only, and they are so constituted that there is neither dilatation nor rotation. On account of them the final formulae are less simple than those of Fresnel. If we suppose the densities to be the same in the two media, there is no correspondence whatever between theory and observation. In this case, as we have seen, vibrations perpendicular to the plane of incidence are reflected according to Fresnel’s tangent-formula; and thus vibrations in the plane of incidence should follow the sine-formula. The actual result of theory is, however, quite different. In the case where the relative index does not differ greatly from unity, polarizing angles of 22½ o and 67½° are indicated, a result totally at variance with observation. As in the case of diffraction by small particles, an elastic solid theory, in which the densities in various media are supposed to be equal, is inadmissible. If, on the other hand, following Green, we regard the rigidities as equal, we get results in better agreement with observation. To a first approximation indeed (when the refraction is small) Green’s formula coincides with Fresnel’s tangent-formula; so that light vibrating in the plane of incidence is reflected according to this law, and light vibrating in the perpendicular plane according to the sine-formula. The vibrations are accordingly perpendicular to the plane of polarization.

The deviations from the tangent-formula, indicated by theory when the refraction is not very small, are of the same general character as those observed by Jamin, but of much larger amount. The minimum reflexion at the surface of glass (μ = 3 / 2 ) would be 1 / 49 , nearly the half of that which takes place at perpendicular incidence, and very much in excess of the truth. This theory cannot therefore be considered satisfactory as it stands, and various suggestions have been made for its improvement. The only variations from Green’s suppositions admissible in strict harmony with an elastic solid theory is to suppose that the transition from one medium to the other is gradual instead of abrupt, that is, that the transitional layer is of thickness comparable with the wave-length. This modification would be of more service to a theory which gave Fresnel’s tangent-formula as the result of a sudden transition than to one in which the deviations from that formula are already too great.

It seems doubtful whether there is much to be gained by further discussion upon this subject, in view of the failure of the elastic solid theory to deal with double refraction. The deviations from Fresnel’s formulae for reflexion are comparatively small; and the whole problem of reflexion is so much concerned with the condition of things at the interface of two media, about which we know little, that valuable guidance can hardly be expected from this quarter. It is desirable to bear constantly in mind that reflexion depends entirely upon an approach to discontinuity in the properties of the medium. If the thickness of the transitional layer amounted to a few wave-lengths, there would be no sensible reflexion at all.

Another point may here be mentioned. Our theories of reflexion take no account of the fact that one at least of the media is dispersive. The example of a stretched string, executing transverse vibrations, and composed of two parts, one of which in virtue of stiffness possesses in some degree the dispersive property, shows that the boundary conditions upon which reflexion depends are thereby modified. We may thus expect a finite reflexion at the interface of two media, if the dispersive powers are different, even though the indices be absolutely the same for the waves under consideration, in which case there is no refraction. But a knowledge of the dispersive properties of the media is not sufficient to determine the reflexion without recourse to hypothesis. xiii. (1871); Von der Miihll, “Ueber die Reflexion und Brechung des Lichtes an der Grenze unkrystallinischen Medien,” Math. Ann., v. 470 (1872), and “Ueber Greens Theorie der Reflexion und Brechung des Lichtes,” Math. Ann., xxvii. 506(1886); Thomson, Baltimore Lectures·, Glazebrook, “Report on Optical Theories,” Brit. Ass. Rep., 1886; Rayleigh, “On Reflection of Vibrations at the Confines of Two Media between which the Transition is gradual,” Proc. Math. Soc., xi.∙, and Walker, “An Account of Cauchy’s Theory of Reflection and Refraction of Light,” Phil. Mag., 151 (1887). References to recent German writers, Ketteler, Lommel, Voigt, &c., will be found in Glazebrook’s Report. ]

§ 28. The Velocity of Light.

According to the principles of the wave theory, the dispersion of refraction can only be explained as due to a variation of velocity with wave-length or period. In aerial vibrations, and in those propagated through an elastic solid, there is no such variation; and so the existence of dispersion was at one time considered to be a serious objection to the wave theory. Dispersion in vacuo would indeed present some difficulty, or at least force upon us views which at present seem unlikely as to the constitution of free aether. The weight of the evidence is, however, against the existence of dispersion in vacuo. “Were there a difference of one hour in the times of the blue and red rays reaching us from Algol, this star would show a well-marked coloration in its phases of increase and decrease. No trace of coloration having been noticed, the difference of times cannot exceed a fraction of an hour. It is not at all probable that the parallax of this star amounts to one-tenth of a second, so that its distance, probably, exceeds two million radii of the earth’s orbit, and the time which is required for its light to reach us probably exceeds thirty years, or a quarter of a million hours. It is therefore difficult to see how there can be a difference as great as four parts in a million between the velocities of light coming from near the two ends of the bright part of the spectrum.”

For the velocity of light in vacuo, as determined in kilometres per second by terrestrial methods (Light, vol xiv. p. 585), Newcomb gives the following tabular statement :— [table]

To these may be added, for reference— [table]

Newcomb concludes, as the most probable result—

Velocity of light in νacuο= 299,860±30 kilometres.

It should be mentioned that Young and Forbes inferred from their observations a difference of velocities of blue and red light amounting to about 2 per cent., but that neither Michelson nor Newcomb, using Foucault’s method, could detect any trace of such a difference.

When we come to consider the propagation of light through ponderable media, there seems to be little reason for expecting to find the velocity independent of wave-length. The interaction of matter and aether may well give rise to such a degree of complication that the differential equation expressing the vibrations shall contain more than one constant. The law of constant velocity is a special property of certain very simple media. Even in the case of a stretched string, vibrating transversely, the velocity becomes a function of wave-length as soon as we admit the existence of finite stiffness.

As regards the law of dispersion, a formula, derived by Cauchy from theoretical considerations, was at one time generally accepted. According to this,

μ = A+Bλ -2 + Cλ -4 + (1); and there is no doubt that even the first two terms give a good representation of the truth in media not very dispersive, and over the more luminous portion of the spectrum. A formula of this kind treats dispersion as due to the smallness of wave-lengths, giving a definite limit to refraction (A) when the wave-length is very large. Recent investigations by Langley on the law of dispersion for rock-salt in the ultra red region of the spectrum are not very favourable to this idea. The phenomena of abnormal dispersion indicate a close connexion between refraction and absorption, and Helmholtz has formulated a general theory of dispersion based upon the hypothesis that it may be connected with an absorbing influence operative upon invisible portions of the spectrum. Upon this subject, which is as yet little understood, the reader may consult Glazebrook’s “Report on Optical Theories.”

The limits of this article do not permit the consideration of the more speculative parts of our subject. We will conclude by calling attention to two recent experimental researches by Michelson, the results of which cannot fail to give valuable guidance to optical theorists. The first of these was a repetition under improved conditions of a remarkable experiment of Fizeau, by which it is proved that when light is propagated through water, itself in rapid movement in the direction of the ray, the velocity is indeed influenced, but not to the full extent of the velocity of the water (v). Within the limits of experimental error, the velocity agrees with a formula suggested by Fizeau on the basis of certain views of Fresnel, viz.,

V = V 0 ±⅛u (2), h

V 0 being the velocity when the medium is stationary. In the case of water, (μ 2 -1)∕μ 2 =⋅437. Conformably with (2), a similar experiment upon air, moving at a velocity of 25 metres per second, gave no distinct effect.

From the result of the experiments upon water we should be tempted to infer that at the surface of the earth, moving through space, the gether still retains what must be coarsely called relative motion. Nevertheless, the second research above alluded to appears to negative this conclusion, and to prove that, at any rate within the walls of a building, the gether must be regarded as fully partaking in the motion of material bodies.