471 



rXIU'LAToRY THKoltY <>K I.KiUT. 



t'Xnt-LATORY THEORY OF LIGHT. 



471 



ThU mode of conceiving of redaction and refraction shows at once 

 that if the point of incidence be slightly varied, and ray starting from 

 a point in tli- in. i. lent, and reaching a point in the reflected or refracted 

 ray, be supposed to follow the varied instead of the actual course, tra- 

 velling in each medium with the velocity appropriate to it, the time of 

 transit will be ultimately unchanged. Thia law may be readily extended 

 to any number of reflection* and refraction*. In a great many cases 

 the time of passage i> leu along the actual than along the varied courae, 

 in which case the proposition becomes equivalent to Format's law of 

 swiftest propagation. The time may, however, be a maximum in 

 place of a minimum, or neither .1 maximum nor a minimum. Tim*, if 

 a ray emanate from a point A, and after reflection at the point p reach 

 the point B, the plane A p B will be perpendicular to the tangent plane at 

 P, to which A p, B P, will be equally inclined ; and if A, B, be made the 

 foci of a prolate spheroid of revolution, the magnitude of which in 

 increased until it pi mm through the point r, it will there touch the 

 reflecting surface. If now this surface touch the spheroid externally 

 at the point r, the sum A P + P B will be a minimum when the point 

 of incidence is slightly varied along the reflecting surface ; but if it 

 touch the spheroid internally, the sum will be a maximum ; and if it 

 both touch and cut the spheroid, so as to be external to it in some 

 directions from r and internal in others, the sum will be neither a 

 maximum nor a minimum. 



When concentric waves fall on the surface of a concave mirror, the 

 points on the latter which are successively reached by the waves 

 become the centres of spherical reflected waves, the directions of whose 

 motions tend towards the axis of the mirror; and the surfaces which 

 touch all the secondary waves of like phase become those of as many 

 general reflected waves of spherical or approximately spherical forms, 

 having their convexities towards the mirror. These general waves go on 

 contracting till they pass successively through some point in the axis, the 

 form of the mirror being such as to permit the directions of the motions 

 of the reflected waves to concur in a point ; from this point, which ia the 

 focus of the mirror, they afterwords diverge as from a radiant point. 

 It is easy to conceive that the general front of a wave formed by a sur- 

 face which touches the secondary waves of like phase refracted 

 in a transparent medium (at a convex surface, for example) may be 

 spherical, and have its convexity towards the refracting surface. 

 These waves will go on contracting, and pass successively through 

 some point in the axis, provided the form of the surface of the 

 medium be such as to permit the directions to concur in one point. 

 This point is the focus, and from it, as from a radiant point, the con- 

 centric waves afterwards diverge. 



In order that the relation between the sines of incidence and refrac- 

 tion may be conformable to the results of experiment, it is necessary 

 to assume that the velocity of the waves is diminished when they enter 

 a medium more dense than that in which they previously moved ; and 

 in this circumstance the undulatory theory is opposed to the theory of 

 emission ; for in the latter the velocity of light is supposed to be 

 increased when it passes from any medium into one more dense. This 

 led Arago to an ap|>arently crucial experiment, to decide between the 

 two theories. If a thin plate of mica be interposed in the path of one 

 of two streams of light proceeding to interfere, the effect, according to 

 the theory of emissions, will be to accelerate, according to that of 

 undulations to retard, the stream passing through it. The direction 

 in which the fringes ore shifted, shows that the effect of the plate is 

 the tame as that of incmuiny the length of path of the stream passing 

 through it, in accordance with the theory of undulations, and in direct 

 contradiction, as it would appear, to the theory of emissions. However, 

 as the decision is only arrived at by referring to another optical effect, 

 depending for its explanation on the view we take of the nature of 

 light, it is satisfactory to be able to refer to Foucault's celebrated 

 experiment, mentioned at the commencement of this article, in which 

 the same result is obtained by direct experiment. (See ' Annales de 

 Chimie,' torn. 41, 1854, p. 129.) 



From the demonstration of the law of refraction according to the 

 undulatory theory, it follows that if ft be the refractive index of a sub- 

 stance, r : r" : : ft : 1. Now, for a given substance ft depends UJHIII the 

 kind of light, increasing, though by no great fraction of the whole, in 

 pwning from the red to the violet According to one of our funda- 

 mental suppositions, colour depends on the periodic time of the vibra- 

 tions, and therefore we are obliged to suppose that one at least of the 

 two velocities of propagation r, v', changes with the periodic time. 

 This has to some appeared a formidable difficulty, inasmuch as theory 

 and experiment combine in showing that musical notes of all degrees 

 of pitch arr ]>ro|>agato<l, in air with the same velocity, and calculation 

 hows that this independence of velocity of propagation and periodic 

 time must hold good in any homogeneous elastic mi!<lium in whi. h 

 undulations are propagated by virtue of the pressures or tensions called 

 into play by the relative displacements within an iml.-finit.-ly small 

 element of the medium nurrounding the point at which the prowure in 

 estimated. To us the objection, even j.ri'.r to the e..n*i. I. -ration of the 

 mode in which the result may be accounted for, does not appear to be 

 at all i.f this formidable character ; for all the phenomena which bear 

 on the subject conspire, as we have seen, to show that tl,- 

 propagation . in vacuo is the same for all colours, and therefore it is 

 to a variation in r" that we are to look to account for the observed 

 variation of /*. Now, according to our supposition, light is propagated 



within water, glass, Ac., by the vibration of the ether within them. 

 But the motion of one of two mutually interpenetrating media is so 

 utterly different from anything we have to deal with in the theory of 

 sound that we cannot reason from the one case to the other. 



But further, a plausible mode of accounting for .li.|-r.-i..u on the 

 undulatory theory has been suggested, which not only removes the 

 objection arising from the existence of a phenomenon which to some 

 might appear inexplicable, but ban led to the discovery of thu ] 

 mate law of dispersion. Freanel, in his memoir on ooobli 

 refers to a note, to apjiear at the end of the memoir, in win. h 

 plains dispersion by supposing that the forces by whn !. 

 particles act on one another ore sensible to a distance which is not inn - 

 nitely small compared with the length of a wave. This appear* 

 by no means a violent supposition to make, when we >:.,( i the 

 extreme smallness of A. The note appears to have been lost, at least 

 it does not accompany the memoir, but the subject was taken u|> by 

 duchy, who has thus been led to express the square of the refractive 

 index by a series according to inverse even powers of A, the wave- 

 length in vacuo. Restricting ourselves to the most important t.-im, 

 we thus get, by ordinary algebraic expansion, ft = A + BA- J , A and 11 

 being constants depending on the nature of the medium, according to 

 which expression A /t ought to vary as A A~ 5 . And that this expression 

 is no mere formula of interpolation, but contains a law of natm 

 one may readily convince himself by taking Fraunhofer's indii 

 some kind of glass, and the wave-lengths of the fixed lines i. n, r. i .... u. 

 as determined by him by a diamond-ruled grating, and subtr.icting t he 

 logarithms of A A- J from those of Aft, A denoting the increments 

 belonging to intervals such as c to o, D to E, c to K, fox, and then 

 comparing the results with those obtained from a formula of interpola- 

 tion taken at random, such as ft'= A -I- B A-', or ft = A + B A," 3 , similarly 

 treated. The constancy of the differences of the logarithm* in all the 

 less refrangible part of the spectrum when the formula resulting from 

 Cauchy's theory is used, cannot leave a moment's hesitation tli.it the 

 formula expresses a natural law. But if, accepting the formula as a 

 true first approximation, we endeavour to ascend from it to the physical 

 circumstances giving rise to it, we see that it merely indie iu-* the 

 existence, in the partial differential equation of motion, of a dill'eivntial 

 coefficient of the fourth order, without even completely specifying the 

 variable or variables with respect to which the differentiation is taken. 

 A result of such generality might well be obtained from a variety of 

 physical hypotheses, so that we must not lay undue stress on the 

 experimental verification of Cauchy's law in considering the ivi.hnro 

 in favour of the physical theory from which he deduced it. In.i.-<l, 

 the fact, as it appears to be, of the absence of a chromatic vari.-n 

 the velocity of propagation in vacuo, would seem to indicate that the 

 molecules of ponderable matter play a very direct jiart in the phe- 

 nomenon of dispersion. 



Hitherto we have confined ourselves to the laws of reflection and 

 refraction, which were in fact demonstrated by Huygens long 1 

 the principle of interference was known. Our subject naturally leads 

 us, in the next place, to interference, to which a special article has 

 already been devoted. [!NTKUFKUKNCK.] In that article the subject 

 has been generally explained, and two fundamental experiments, due 

 to Fresnel, have been mentioned, which show that interference is an 

 essential property of light. We shall here, therefore, proceed t 

 formula which gives the intensify of the light resulting from the 

 mixture of two interfering streams. 



It will be necessary, in the first place, to express analytically the 

 disturbance in a single stream of light. We shall suppose, for the 

 sake of simplicity, that the waves are plane, and that the maximum 

 accession of the particles of ether is the same at one point of space as 

 another. ThU will be sufficient in any cose, provided we confine ..in- 

 attention to a small portion only of the fronts of the waves, an.l to 

 variations of distance in a direction perpendicular to the from. 

 that the change of intensity due to convergence or divergence nceil not 

 be taken into account. Let the ether be referred to the rectangular 

 axes of x, y, z, x being measured in the direction of propagation ; let ( 

 be the time, and r the velocity of propagation. Since the waves are 

 supposed plane, the disturbance will be independent of y and 

 therefore will depend only on ,r and (. Moreover, according to the 

 fundamental notion of an undulation, whatever ili.-turKancc exist* at 

 the time t at the distance x from the origin will, after the lapse of the 

 time 8 (, be found at a plane further in advance by S .r, 8 being con- 

 nected with S t by the relation 5.i = uSf. Hence the . 

 remains the same, provided the difference between rl ami .; remains 

 unchanged, and therefore for one value of r ( x the disturbance will 

 have one value, for another value another, and so on. In other \, 

 the disturbance will be some function ^ (v t x) of v t x, the direc- 

 tion i.f the motion of the particles being left a perfectly open que 

 The character of the undulations will depend on the form f t he func- 

 tion 4.. Now we have seen reason to believe that in light we always 

 have to deal with a succession of a great number of similar p''ii.ii<- 

 disturbances, ami, further, that the colour depends upon the p 

 time. But in all optical phenomena the effects of lights of dill 

 colours are simply superposed, and therefore as regards the form of \fi 

 we are justified in restricting ourselves to the consideration of a 

 regularly periodic function. Among such functions there is one v 

 claims our special attention, namely, that which is expressed by a sinu 



