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As a = B, we see that when there is no transmitted wave the intensity of the reflected wave is precisely equal to that of the incident one. This is what might be expected : it is, however, noticed here because a most illustrious analyst has obtained a different result. (Poisson, Mémoires de l'Academie des Sciences, Tome x.) The result which this celebrated mathematician arrives at is, That at the moment the transmitted wave ceases to exist, the intensity of the reflected becomes precisely equal to that of the incident wave. On increasing the angle of incidence this intensity again diminishes, until it vanish at a certain angle. On still farther increasing this angle the intensity continues to increase, and again becomes equal to that of the incident wave, when the angle of incidence becomes a right angle.
It may not be altogether uninteresting to examine the nature of the disturbance excited in that medium which has ceased to transmit a wave in the regular way. For this purpose, we will resume the expression,
0,= Bens sid y = Bemua sin (by + ct); or if we substitute for B, its value given by the last of the equations (10); and for a', its value from (11); this expression, in the case of ordinary elastic fluids where yA=, ,, will reduce to
0,=2ajo cose. Tu s in (by + ct), à being the length of the incident wave measured perpendicular to its own front, and the angle of incidence. We thus see with what rapidity in the case of light, the disturbance diminishes as the depth x below the surface of separation of the two media increases; and also that the rate of diminution becomes less as 6 approaches the critical angle, and entirely ceases when @ is exactly equal to this angle, and the transmission of a wave in the ordinary way becomes possible.
ON THE LAWS
REFLEXION AND REFRACTION OF LIGHT
AT THE COMMON SURFACE OF TWO NON..
From the Transactions of the Cambridge Philosophical Society, 1838.
ON THE LAWS OF THE REFLEXION AND REFRACTION
OF LIGHT AT THE COMMON SURFACE OF TWO
M. CAUCHY seems to have been the first who saw fully the utility of applying to the Theory of Light those formulæ which represent the motions of a system of molecules acting on each other by mutually attractive and repulsive forces ; supposing always that in the mutual action of any two particles, the particles may be regarded as points animated by forces directed along the right line which joins them. This last supposition, if applied to those compound particles, at least, which are separable by mechanical division, seems rather restrictive; as many phenomena, those of crystallization for instance, seem to indicate certain polarities in these particles. If, however, this were not the case, we are so perfectly ignorant of the mode of action of the elements of the luminiferous ether on each other, that it would seem a safer method to take some general physical principle as the besis of our reasoning, rather than assume certain modes of action, which, after all, may be widely different from the mechanism employed by nature; more especially if this principle include in itseli, as a particular case, those before used by M. Cauciy and others, and also lead to a much more simple process of calculation. The principle selected as the basis of the reasoning contained in the following paper is this : In whatever way the elements of any material system may act upon each other, if all the internal forces exerted be multiplied by the elements of their respective directions, the total sum for any assigned portion of the mass will always be the exact differential of some function. But, this function being known, we can immediately apply the general method given in the Mécanique Analytique, and which appears to be more especially applicable to
given in the Mécanique Anulytique became abundantly sufficient for the solution of the problem.
In conclusion, it may be observed, that the radius of the sphere of sensible action of the molecular forces has been regarded as insensible with respect to the length a of a wave of light, and thus, for the sake of simplicity, certain terms have been disregarded on which the different refrangibility of differently coloured rays might be supposed to depend. These terms, which are necessary to be considered when we are treating of the dispersion, serve only to render our formulæ uselessly complex in other investigations respecting the phenomena of light
Let us conceive a mass composed of an immense number of molecules, acting on each other by any kind of molecular forces, but which are sensible only at insensible distances, and let moreover the whole system be quite free from all extraneous action of every kind. Then x, y and z being the co-ordinates of any particle of the medium under consideration when in equilibrium, and
*+u, y+v, %+ w, the co-ordinates of the same particle in a state of motion (where U, v, and w are very small functions. of the original co-ordinates (x, y, z), of any particle and of the time (t)), we get, by combining D'Alembert's principle with that of virtual velocities,
Dm, and Du being exceedingly small corresponding elements of the mass and volume of the medium, but which nevertheless contain a very great number of molecules, and do the exact differential of some function and entirely due to the internal actions of the particles of the medium on each other. Indeed, if 80 were not an exact differential, a perpetual motion would be possible, and we have every reason to think, that