the forces in nature are so disposed as to render this a natural impossibility. Let us now take any element of the medium, rectangular in a state of repose, and of which the sides are da, dy, dz; the length of the sides composed of the same particles will in a state of motion become where dx (1+8,), dy = dy (1+8,), dz' = dz (1 +8 ̧) ; 81, 82, 8, are exceedingly small quantities of the first order. If, moreover, we make, a, B, and y will be very small quantities of the same order. But, whatever may be the nature of the internal actions, if we represent by 84 dx dy dz, the part of the second member of the equation (1), due to the molecules in the element under consideration, it is evident, that will remain the same when all the sides and all the angles of the parallelopiped, whose sides are dx dy dz′, remain unaltered, and therefore its most general value must be of the form $ = function {s,, 82, 83, A, B, Y}. But s1, 82, 83, α, B, y being very small quantities of the first order, we may expand in a very convergent series of the form 40, 41, 41, &c. being homogeneous functions of the six quantities a, B, y, s1, 5, 8, of the degrees 0, 1, 2, &c. each of which is very great compared with the next following one. If now, p represent the primitive density of the element dx dy dz, we may write p dx dy dz in the place of Dm in the formula (1), which will thus become, since , is constant, problems that relate to the motions of systems composed of an immense number of particles mutually acting upon each other. One of the advantages of this method, of great importance, is, that we are necessarily led by the mere process of the calculation, and with little care on our part, to all the equations and conditions which are requisite and sufficient for the complete solution of any problem to which it may be applied. The present communication is confined almost entirely to the consideration of non-crystallized media; for which it is proved, that the function due to the molecular actions, in its most general form, contains only two arbitrary coefficients, A and B; the values of which depend of course on the unknown internal constitution of the medium under consideration, and it would be easy to shew, for the most general case, that any arbitrary disturbance, excited in a very small portion of the medium, would in general give rise to two spherical waves, one propagated entirely by normal, the other entirely by transverse, vibrations, and such that if the velocity of transmission of the former wave be represented by A, that of the latter would be represented by B. But in the transmission of light through a prism, though the wave which is propagated by normal vibrations were incapable itself of affecting the eye, yet it would be capable of giving rise to an ordinary wave of light propagated by transverse vibrations, except in the extreme cases where or A B A B = 0, = a very large quantity; which, for the sake of simplicity, may be regarded as infinite; and it is not difficult to prove that the equilibrium of our medium would be unstable unless A 4 B We are therefore compelled to adopt the latter value of A В B' 3 and thus to admit that in the luminiferous ether, the velocity of transmission of waves propagated by normal vibrations is very great compared with that of ordinary light. The principal results obtained in this paper relate to the tensity of the wave reflected at the common surface of two media, both for light polarized in and perpendicular to the plane of incidence; and likewise to the change of phase which takes place when the reflexion becomes total. In the former case, our values agree precisely with those given by Fresnel; supposing, as he has done, that the direction of the actual motion of the particles of the luminiferous ether is perpendicular to the plane of polarization. But it results from our formulæ, when the light is polarized perpendicular to the plane of incidence, that the expressions given by Fresnel are only very near approximations; and that the intensity of the reflected wave will never become absolutely null, but only attain a minimum value; which, in the case of reflexion from water at the proper angle, is part of that of the incident wave. This minimum value increases rapidly, as the index of refraction increases, and thus the quantity of light reflected at the polarizing angle, becomes considerable for highly refracting substances, a fact which has been long known to experimental philosophers. It may be proper to observe, that M. Cauchy (Bulletin des Sciences, 1830) has given a method of determining the intensity of the waves reflected at the common surface of two media. He has since stated, (Nouveaux Exercises des Mathématiques,) that the hypothesis employed on that occasion is inadmissible, and has promised in a future memoir, to give a new mechanical principle applicable to this and other questions; but I have not been able to learn whether such a memoir has yet appeared. The first method consisted in satisfying a part, and only a part, of the conditions belonging to the surface of junction, and the consideration of the waves propagated by normal vibrations was wholly overlooked, though it is easy to perceive, that in general waves of this kind must necessarily be produced when the incident wave is polarized perpendicular to the plane of incidence, in consequence of the incident and refracted waves being in different planes. Indeed, without introducing the consideration of these last waves, it is impossible to satisfy the whole of the conditions due to the surface of junction of the two media. But when this consideration is introduced, the whole of the conditions may be satisfied, and the principles given in the Mécanique Analytique 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 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 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 De being exceedingly small corresponding elements of the mass and volume of the medium, but which neverthe less 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 do were not an exact differential, a perpetual motion would be possible, and we have every reason to think, that the forces in nature are so disposed as to render this a natural impossibility. Let us now take any element of the medium, rectangular in a state of repose, and of which the sides are dr, dy, dz; the length of the sides composed of the same particles will in a state of motion become dxdx (1+8), dy = dy (1+8,), dz' = dz (1+8); where §, 8, 8, are exccedingly small quantities of the first order. If, moreover, we make, a, B, and y will be very small quantities of the same order. But, whatever may be the nature of the internal actions, if we represent by 84 dx dy dz, the part of the second member of the equation (1), due to the molecules in the clement under consideration, it is evident, that will remain the same when all the sides and all the angles of the parallelopiped, whose sides are dx dy dz', remain unaltered, and therefore its most general value must be of the form $= function {s1, 82, 83, a, B, v}. But s1, 82, 83, 4, B, y being very small quantities of the first order, we may expand in a very convergent series of the form ,,,,, &c. being homogeneous functions of the six quantities a, B, y, 81, 8, 8, of the degrees 0, 1, 2, &c. each of which is very great compared with the next following one. If now, p represent the primitive density of the element dx dy dz, we may write p da dy dz in the place of Dm in the formula (1), which will thus become, since p, is constant, |