When concentric waves fall on the surface of a concave mirror, the points on the latter which are euccessively 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 is the focus of the mirror, they afterwards 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 he 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 apparently 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 are shifted, shows that the effect of the plate is the same as that of increasing 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 do Chitnie,' tom- 41, 1854, p. 120.) From the demonstration of the law of refraction according to the undulatory theory, it follows that if e be the refractive index of a sub stance, r : r': : 1. Now, for a given substance e depends upon the kind of light, increasing, though by no great fraction of the whole, in passing 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 arc obliged to suppose that one at least of the two velocities of propagation r, r', 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 are propagated, in air with the same velocity, and calculation shows that this independence of velocity of propagation and periodic time must hold good in any homogeneous elastic medium in which undulations; are propagated by virtue of the pressures or tensions called into play by the relative displacements within an indefinitely small element of the medium surrounding the point at which the protium is estimated. To us the objection, even prior to the consideration of the
mode in which the result may be accounted for, does not appear to be at all of this formidable character ; for all the phenomena which bear on the subject conspire, as we have seen, to show that the velocity of propagation ro,in vacuo is the same for all colours, and therefore it is to a variation in s' that we are to look to account for the observed variation of n. Now, according to our supposition, light is propagated within water, glass, &c., 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 dispersion on the undulatory theory hart been suggested, which not only removes the objection arising from the existence of a phenomenon which to some might appear inexplicable, but has led to the discovery of the approai. mate law of dispersion. Frame!, in his memoir on double refraction, refers to a note, to appear at the end of the memoir, in he ex plains dispersion by supposing that the forms by which the ethereal 'articles act on one another are sensible to a distance which is not infi nitely small compared with the length of a wave. This appears to be by no means a violent supposition to make, when we consider the extreme smallness of a. The note appears to have been lost, at least it does not acconipany the memoir, but the subject was taken up by Cauchy, 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 term, we thus get, by ordinary algebraic expansion, ti = A + n A and being constants depending on the nature of the medium, according to which expression A e ought to vary as a And that this expression is no mere formula of interpolation, but contains a Lew of nature, any one may readily convince Mined! by taking Fraunhofer's indices for some kind of glass, and the wave-lengths of the fixed lines c, e, r, as determined by him by a diamond-ruled grating, and subtracting the logarithms of A A"' from those of A IA, a denoting the increments belonging to intervals such as c to n, n to E, c to E, &c., and then comparing the results with those obtained from a formula of interpola tion taken at random, such as = A + n or ti=A-1-n similarly treated. The constancy of the differences of the logarithms in all the less refrangible part of spectrum when the formula resulting from Cauchy'e theory is used, cannot leave a moment's hesitation that 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 indicates the existence, iu the partial differential equation of motion, of a differential 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 evidence in favour of the physical theory from which he deduced it. Indeed, the fact, as it appears to be, of the absence of a chromatic variation of the velocity of propagation in vacua, would seem to indicate that the molecules of ponderable matter play a very direct part in the phe nomenon of dispersion.