() and canceling the common factor , we obtain This equation was derived by Augustin-Jean Fresnel in January 1822. If is the length of the radius vector, the equation describes a surface with the property that the major and minor semiaxes of any diametral section have lengths equal to the wave-normal speeds of wavefronts parallel to that section, and the directions of what Fresnel called the "vibrations" (which we now recognize as oscillations of ). Whereas the surface described by () is in index space (in which the coordinates are dimensionless numbers), the surface described by () is in velocity space (in which the coordinates have the units of velocity). Whereas the former surface is of the 2nd degree, the latter is of the 4th degree, as may be verified by redefining as the components of velocity and putting , etc; thus the latter surface () is generally not an ellipsoid, but another sort of ovaloid. And as the index ellipsoid generates the index surface, so the surface (), by the same process, generates what we call the normal-velocity surface. Hence the surface () might reasonably be called the "normal-velocity ovaloid". Fresnel, however, called it the surface of elasticity, because he derived it by supposing that light waves were transverse elastic waves, that the medium had three perpendicular directions in which a displacement of a molecule produced a restoring force in exactly the opposite direction, and that the restoring force due to a vector sum of displacements was the vector sum of the restoring forces due to the separate displacements. Fresnel soon realized that the ellipsoid constructed on the same principal semi-axes as the surface of elasticity has the same relation to the ray velocities that the surface of elasticity has to the wave-normal velocities. Fresnel's ellipsoid is now called the ray ellipsoid. Thus, in modern terms, the ray ellipsoid generates the ray velocities as the index ellipsoid generates the refractive indices. The major and minor semiaxes of the diametral section of the ray ellipsoid are in the permitted directions of the electric field vector . The term index surface was coined by James MacCullagh in 1837. In a previous paper, read in 1833, MacCullagh had called this surface the "surface of refraction" and shown that it is generated by the major and minor semiaxes of a diametral section of an ellipsoid which has principal semiaxes inversely proportional to those