UOTULATORY THEORY OF LIGHT. 



UNDULATORY THEORY OF LIGHT. 



MO 



of reflection!, u Poisson bad previously done in the owe of a ptrpen 

 dicuUr incidence, that the minima in the reflected rings ought at any 

 incidence to be perfectly black, and that, without assuming anything 

 relative to the law of intensity in reflection beyond a law discovered 

 experimentally by Arago, that at any obliquity light i reflected in 

 the tame proportion at the first and second surfaces of a transparent 

 plate. Kur a very simple demonstration at the same time of Arago's 

 law, and of the loss of a half undulation, on the assumption merely 

 that the forces acting depend only on the positions of the particles, the 

 reader is referred to a paper in the ' Cambridge and Dublin Mathe- 

 matical Journal ' (vol. iv. p. 1). A complete investigation of the 

 intensities of the reflected and transmitted rings, in which account is 

 taken of the infinite number of reflections, will be found in Airy's 

 Tract,' arts. 64-7. The result U 



Fur the reflected system 



4aVnn*-Y 

 (1 -()> + 4 ( sin* f v 



- 

 For the transmitted system , 



(!-) + 4e' sin 5 -v 



where a denotes the coefficient of vibration for the incident light, 

 v tin' retardation, 2 ncos/3, and 1 : r the ratio in which the coefficient 

 of vibration U altered by one reflection. The sum of the two expres- 

 sions is always equal to of, which shows that the reflected and refracted 

 systems are complementary to each other, conformably to observation. 



Sir Isaac Newton also discovered that when the sun's light is 

 reflected into a darkened room, and allowed to fall on a screen with a 

 moderately small hole, and the beam so passing is received perpendicu- 

 larly on a concave mirror, made of glass quicksilvered at the back, 

 which is placed at such a distance from the screen that the hole coin- 

 cides with the centre of curvature of its surfaces, so that the regularly 

 reflected light goes bock through the same small hole by whirl, it 

 entered, a system of coloured rings is seen depicted on the screen, on 

 the face towards the mirror, surrounding the hole. The order of 

 colours, and the law of the diameters of the rings, agree with the 

 transmitted system of the rings formed between two object-glasses. 

 A metallic K|ieculum exhibits no such rings. If the amalgam be 

 removed from a mirror of quicksilvered glass, the rings ore seen as 

 before, but much fainter. With different mirrors the diameter of a 

 ring varies directly as the radius of curvature of the surfaces, 

 and inversely as the square root of the thickness of the glass. 



Although Newton expressly refers to the defect of polish of the first 

 surface in relation to these rings, he does not seem to hare purposely 

 tarnished his mirrors. In repeating Newton's experiments, the Duke 

 de Chaulnes observed that the brilliancy of the rings was very greatly 

 increased by tarnishing the surface, for which milk much diluted with 

 water is very convenient. It is advantageous to form a diverging beam 

 by transmitting a beam of sunlight through a pretty large lens, at the 

 focus of which is to be placed the small hole of the screen. In this 

 way of operating the experiment U one of remarkable beauty. On 

 slightly inclining the mirror, the phenomenon changes in a very 

 remarkable way, a set of coloured rings continually opening out from a 

 point midway between the luminous point, or image of the sun in the 

 focus of the lens, and its image formed by regular reflection. The 

 experiment may be varied in a very beautiful way by dispensing with 

 screen and sunlight, and simply placing a small taper-flame in front of 

 the tarnished mirror, in such a position as to coincide with its inverted 

 image. On viewing the coincident flame and image, they are seen 

 surrounded by a splendid series of coloured rings, which appear to 

 have a determinate position in the air like on actual object 



These phenomena are known as the calourt </ thick plates. They 

 have been shown to arise from the interference of two streams of 

 whereof one U scattered on entering the glow, and then regu- 

 larly reflected at the back and refracted out ; and the other enters the 

 glass by regular refraction, and after regular reflection at the back is 

 scattered in emerging. At a point coinciding with the luminoun point 

 and ita image, the two scattered streams follow the course of the 

 regular light, and therefore their difference of path is nothing, and the 

 difference of path of the two scattered streams, w Inch reach a point at 

 no great distance from the former, will accordingly be .-m' 

 small to allow the streams to manifest the ordinary phenomena <"f 

 interference. The theory of the rings for the most important case, 

 that in which the luminous point is In the centre of curvature of the 

 mirror, U given in Sir John Herschel's 'Treatise on Light,' arta. 

 679, Ac. 



1 >r. Whcwell and M. Quetelet observed a system of coloured bonds, 

 which are seen when the flame of a candle held near the eye is viewed 

 by reflection in a common looking-glass, several feet off, with a tar- 

 nished surface. To see them distinctly, the image of the flame must 

 be seen distinctly, so that an eye-glass must he used if n-<|uii. .1. Ti,, ; , 

 change with every change of position of the candle or of the eye. ami 

 with both eyes a double system is seen, one with each eye. Their 

 explanation depends on the same principles as that of the ring* : 

 by concave mirrors of quicksilvered glass, and the theory of both kind* 



will be found treated with great detail in a paper published in the 

 ' Cambridge Philosophical Transactions,' vol. ix., part 2, p. 147. 



The subject of diffraction has been already briefly considered in a 

 special article. [DIFFRACTION or LIGHT.] The full comparison of 

 theory and experiment with reference to this curious and interesting 

 class of phenomena, requires far too much mathematical calculation 

 to be here introduced, and reference must be made to Airy's Tract 

 on the Undulatory Theory, or other works in which the question i* 

 treated. Suffice it to remark, that the undulatory theory has led to a 

 most complete explanation of the phenomena, qualitatively and quan- 

 titatively, in their minutest details. 



There is one phenomenon, the astronomical phenomenon of aberra- 

 tion, the explanation of which on the corpuscular theory is so simple as 

 to attract little notice, but which presents a serious difficulty on the 

 theory of undulations. To account fur it, it has been supposed by 

 Dr. Young and others, that the earth in ita motion round the sun 

 passes through the ether without disturbing it, allowing it to pan 

 between ita own particles like the wind through a grove of trees. 

 Startling as this hypothesis is, we ought not to reject it on the strength 

 merely of previous notions respecting such a mysteriously subtle 

 medium as the luminiferous ether, if we were fairly led to it. But 

 we are not obliged to have recourse to it, for the phenomenon admit* 

 of explanation by attributing to the ether a motion, due to bodies 

 passing through it, which is of a kind with which we have a gre.. 

 to do in hydrodynamics. (See ' Philosophical Magazine,' vol. xxvii. 

 (1845), p. 9, and several subsequent articles.) 



In the explanation of the phenomena which we have hitherto con- 

 sidered, nothing depends upon the direction of vibration of the \>., 

 of ether which transmit the waves, but the phenomena of polarisation 

 lead us to suppose that the vibrations are transverse to the direction of 

 propagation. [POLARISATION OF LIGHT.] This supposition 

 admitted, the curious and complicated phenomena of the interference of 

 polarised light are explained with beautiful simplicity. We l< 

 this article to consider how far the same hypothesis of transverse 

 vibrations helps tie towards an explanation of double refraction. 



\\ .' shall commence by briefly adverting to the facts of double 

 refraction, and to its laws so for as they were ascertained l f">e 

 Freanel's researches on the subject It was in Inland spar that th 

 phenomenon was finst discovered, and this crystal, from the great ] .. >wer 

 of ita double refraction was well suited for a study of the MI 

 especially at a time when the instrumental means uf examination \vru> 

 far inferior to what we at present possess. Of the two ray 

 which Iceland spar divides in general a single ray incident UJHUI 

 more refracted was found to obey the ordinary law of refraction, but 

 the less refracted was found to obey a more complicated law, not >-\<-\\ 

 lying in the plane of incidence, except in particular cases. On 

 measuring the refractive index of the spar with respect to the lat 

 extraordinary ray by methods applicable to ordinary me.1 

 values were obtained, varying from a maximum equal to the refractive 

 index for the ordinary ray, and obtained when the course of the ray 

 within the crystal was parallel to the axis, to a minimum, whi> 

 obtained in any direction perpendicular to the axis, around which 

 everything relating to the optical properties was symmetrical, 

 there are here two rays, we must, if we adopt the undulatory ; 

 at all, assume that a disturbance excited at any point of the < 

 would give rise not to one wave, but to two waves, diverging from 

 that point, or what comes to the same, to a wave represented I >y a 

 surface of two sheets. As the more refracted ray in I 

 the ordinary law of refraction, the inner sheet of this surface must be 

 supposed to be a sphere. The facts which have been mentioned 

 respecting the refraction of the extraordinary ray, show tli.it the outer 

 sheet must be a surface of revolution around the axis of the < r 

 along which it touches the inner sheet, and must be most protuberant 

 at the equator. Apparently as being next in simplicity to a sphere, 

 Huygens assumed this surface or sheet to be an oblate spheroid of 

 revolution, and found the calculations thence resulting as to the 

 course of the extraordinary ray confirmed by the result of hi* 

 experiments. 



The demonstration of the laws of reflection and refraction in the case 

 of ordinary media, require but a slight modification to adapt them to 

 the case of a crystal for which we ore supposed to know the form of 

 the wave surface. In fact, if we suppose, far simplicity, the incident 

 waves and the surface of the crystal to be Imth plain-, we have only to 

 replace the hemispheres within the media by the corresponding wave 

 surfaces, which will form a system of similar ami similarly .- 

 curved surfaces with two plane envelopes, one for each nheet Hence 

 results the following construction. Draw a line perpendicular i 

 incident waves in air, and therefore representing the course of an inci- 

 dent ray. With the point of incidence for centre, describe a Bplieio 

 representing the velocity of propagation in air, and likewise within the 

 refracting medium draw the wave surface on a corresponding scale, so 

 that ita radius in any direction represents the velocity of a ray | 

 .jatid in that direction. Produce the incident ray to i met the spin i. 

 within the refracting medium which in a continuation of the hemi- 

 sphere described in air, and at the point of i tin;- dm 



ilane to the sphere, or, in other words, a plane perpendicular to I he 

 ray. Through the line of jiitarsectiiin of thin tangent plane with the 

 >lnc of the surface, draw tangent planes to the two sheets respectively 



