sented by equations formed by equating to unity the sums of the squares of the n variables, each divided by an appropriate coefficient. It is of course possible to adapt the formula of this paper to the case of nature by supposing n = 3. The next paper, "On the Motion of Waves in a variable canal of small depth and width," though short, is interesting. It was read before the Cambridge Philosophical Society, on May 15, 1837, and a Supplement to it on Feb. 18, 1839. On Dec. 11, 1837, were communicated two of his most valuable memoirs, "On the Reflexion and Refraction of Sound," and "On the Reflexion and Refraction of Light at the common surface of two non-crystallized media." These two papers should be studied together. The question discussed in the first is, in fact, that of the propagation of normal vibrations through a fluid. Particular attention should be paid to the mode in wbich, from the differential equations of motion, is deduced an explanation of a phenomenon analogous to that known in Optics as Total internal reflection when the angle of incidence exceeds the critical angle. By supposing that there are propagated, in the second medium, vibrations which rapidly diminish in intensity, and become evanescent at sensible distances, the change of phase which accompanies this phenomenon is clearly brought into view. The immediate object of the next paper, "On the Reflexion and Refraction of Light at the common surface of two noncrystalline media," is to do for the theory of light what in the former paper has been done for that of sound. This is done in a manner which will present little difficulty to one who has mastered the former paper. But this paper has an interest extending far beyond this subject. For the purpose of explaining the propagation of transversal vibrations through the luininiferous ether, it becomes necessary to investigate the equations of motion of an elastic solid. It is here that Green for the first time enunciates the principle of the Conservation of work, •which he bases on the assumption of the impossibility of a perpetual motion. This principle he enunciates in the following words: "In whatever manner the elements of any material system may act upon each other, if all the internal forces 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." This function, it will be seen, is what is now known under the name of Potential Energy, and the above principle is in fact equivalent to stating that the sum of the Kinetic and Potential Energies of the system is constant. This function, supposing the displacements so small that powers above the second may be neglected, is shewn for the most general constitution of the medium to involve twenty-one coefficients, which reduce to nine in the case of a medium symmetrical with respect to three rectangular planes, to five in the case of a medium symmetrical around an axis, and to two in the case of an isotropic or uncrystallized medium. The present paper is devoted to the consideration of the propagation of vibrations from one of two media of this nature. The two coefficients above mentioned, called respectively A and B, are shewn to be proportional to the squares of the velocities of propagation of normal and transversal vibrations respectively. It is to be regretted that the statical interpretation is not also given. It may however be shewn (see Thomson and Tait's Natural Philosophy, p. 711 (m.)) that A — fS measures the resistance of the medium to com pression or dilatation, or its elasticity of volume, while B measures its resistance to distortion, or its rigidity. The equilibrium of the medium, it may be shewn, cannot be stable, unless both of these quantities are positive*. A Supplement to this paper supplying certain omissions, immediately follows it. In the next paper, "On the Propagation of Light in Crystalline Media," the principle of Conservation of Work is again assumed as a starting-point and applied to a medium of any description. It is first assumed that the medium is symmetrical with respect to three planes at right angles to one another, by which supposition the twenty-one coefficients previously mentioned are reduced to nine. Fresnel's supposition, that the vibrations affecting the eye are accurately in front of the wave, is then introduced, and a complete explanation of the phenomena of polarization is shewn to follow, on the hypothesis that the vibrations constituting a plane-polarized ray are in the plane of polarization. The hypothesis adopted in the former paper—that these vibrations are perpendicular to the plane of polarization—is then resumed, and an explanation arrived at, by the aid of a subsidiary assumption—unfortunately not of the same simple character as those previously introduced—that for the three principal waves the wave-velocity depends on the direction of the disturbance only, and is independent of the position of the wave's front. The paper concludes by taking the case of a perfectly general medium, and it is shewn that Fresnel's supposition of the vibrations being accurately in the wave-front, gives rise to fourteen relations among the twenty-one coefficients, which virtually reduce the medium to one symmetrical with respect to three planes at right angles to one another. * In comparing Green's paper with the passage in Thomson and Tait's Natural Philosophy above referred to, it should be remarked that the A of the former is equal to the m - \n of the latter, and that B—n. This paper, read May 20, 1839, was his last production. Another, "On the Vibrations of Pendulums in Fluid Media," read before the Royal Society of Edinburgh, on Dec. 16, 1833, will be found at the end of this collection. The problem here considered is that of the motion of an inelastic fluid agitated by the small vibrations of a solid ellipsoid, moving parallel to itself. I have to express my thanks to the Council of the Cambridge Philosophical Society, and to that of the Royal Society of Edinburgh, for the permission to reproduce the papers published in their respective Transactions which they have kindly given. N. M. FERRERS. GONVILLE AND Caius COLLEGE, Dec. 1870. CONTENTS. rui An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism ... ... ... 1 Preface ... ... ... ... ... ... 3 Introductory Observations ... ... ... ... 9 General Preliminary Results... ... ... ... 19 Application of the preceding results to the Theory Application of the preliminary results to the Theory of Mag- netism ... ... ... ... ... ... ... 83 Mathematical Investigations concerning the Laws of the Equi- librium of Fluids analogous to the Electric Fluid, with other similar researches ... ... ... ... ... 117 On the Determination of the Exterior and Interior Attractions of Ellipsoids of Variable Densities ... ... ... 185 On the Motion of Waves in a variable canal of small depth On the Reflexion and Refraction of Sound ... ... ... 231 On the Laws of the Reflexion and Refraction of Light at the common surface of two non-crystallized Media ... ... 243 Note on the Motion of Waves in Canals ... ... ... 271 Supplement to a Memoir on the Reflexion and Refraction On the Propagation of Light in crystallized Media ... ... 291 Researches on the Vibration of Pendulums in Fluid Media ... 313 |