characterized by longitudinal and transverse vibrations, respectively, [Pg 20] are distinguished, and the corresponding wave-velocities found. A great improvement in the theory was made by Cauchy, who initiated the modern theory of stress and strain. As an alternative to the method which he had first adopted, he abandons all explicit mention of molecules, and treats a solid as practically continuous. Extending the notion of pressure which was current in Hydrostatics, he assumes that the force between any two adjacent parts of a substance can be regarded as made up of actions between two strata of excessively small depth on the two sides of the interface, and may accordingly be treated as a surface-force or “stress.” He goes on to investigate the relation between the stresses across different planes, and to express them geometrically by means of the stress-ellipsoid. This use of an ellipsoid to represent the relations between various directional properties in Mechanics is I believe original with Cauchy, who applied it also in the theory of strains, as well as in the more familiar [Pg 21] matter of moments of inertia. His equations for an isotropic substance, obtained by this second method, are based on the hypothesis that the principal axes of stress and strain coincide, and have the now usual form, with two independent elastic constants. The whole procedure is in fact that found in modern books. It should be mentioned also that Cauchy in his work on strains introduces for the first time the notion of the infinitesimal rotation of an element, afterwards utilized by Stokes and Helmholtz. Cauchy next took up the theory of crystalline solids, this time naturally on the basis of an assumed orderly arrangement of molecules, but his results have failed to stand the test of experiment, or to furnish a satisfactory explanation of double-refraction. The true theory of elastic solids in the general case, free from all molecular hypothesis, was given later by Green, whose work is the first example of the application of energy-methods to the physics of continua, the analytical process being an adaptation of the variational [Pg 22] method of Lagrange. It is fortunately not my task to discuss these things from the point of view of Physical Optics, or to review the long-continued and obstinate attempts of successive physicists to construct a mechanical model of the ether, now definitely abandoned. At the present time the real outlet for the theory of elastic waves and their reflection and refraction is in relation to Seismology, where it has led to important results. The chief interest of the theory of Elasticity to us at the moment consists partly in the gradual emancipation from molecular assumptions, and partly in that the analytical relations which it involved were destined to find a wider and more important sphere of application. To take a very simple instance, in the equations of equilibrium of an incompressible isotropic solid,