of the angles which their common direction makes with the optic axes suggested (apparently) that the same Theorem approximately held when wave was put for ray, and normal to front for direction, &c., and thus a Theorem which is in no way connected with the result, does from the circumstance of its close analogy to the true one, give correct results, or nearly so. 3. Let BC be the direction of one ray in the crystal; BE a normal to its front; CG perpendicular to BA; the angle of incidence; p' the angle which BE makes with the normal to the plane surface of the crystal; BC makes with the same; T the thickness of the plate. (Note at end.) Then if be the velocity before incidence, the velocity perpendicular to the front after refraction, and the ray has moved perpendicularly to its former front through a space therefore the space which the wave would describe in the same time in air, is and if v be the velocity of the other wave, its retardation is the angle of emergence being supposed the same for both. Hence the difference of retardation is = Tv T{-} To{-} nearly. || 4. This hypothesis that the two waves are moving parallel to each other at emergence, is clearly not compatible with the hypothesis that they have the same normal within the crystal. If v2 be the velocity of the wave which has a common normal with that whose velocity is v1, we have hence the hypothesis that the angle of incidence is small, reduces this case to the same form as the former, and we may in such circumstances consider the difference of the retardation as proportional to the difference between the two refractive indices. 5. In the applications of this formula, we must introduce the relations which are given by the constitution of the crystal determined by the passage of light through it. Such relations must, I conceive, depend on the refractive energies of the crystal in different directions. Now the refractive energy has undoubtedly no connexion whatever with the velocities of transmission of the rays, since these velocities are merely nominal ones; that is, they are not estimated in the direction in which the effect is transmitted. Indeed, I do not suppose we have any notion of these velocities independent of theory, whilst the velocity of the wave is a physical motion, apart from the idea which is suggested by the expression. I have been under the necessity of giving the term radial to M. Fresnel's axes, since they are not at all the same thing as the optic axes. M. Fresnel himself remarks, that "although the difference between them is very slight in almost all crystals, there are some where it becomes more sensible, and in which we must not confound the two." common 6. We are concerned only with waves which have a direction in air, and must consequently assume that the difference of the velocities of the two corresponding refracted waves, is very nearly the same as the difference of the velocities of two waves which travel in the direction of one of them, omitting consequently the variation of velocity of one wave due to difference of its velocity from that of the other, or in other words, omitting the variation of the difference of the velocities, compared with that difference itself which is perfectly allowable. Let m', n' be the angles made by the direction, which we consider common to the two, with the optic axes. a, ẞ the angles made in air by the incident ray, with rays, which, when they enter the crystal, move in rays; the normals to which are the axes. so that sin = μ sin d' as a factor of small terms. 2 sin3a sin2ß = {sin' + sin2 Tz – sin2 sin3T≈ (1 + sin20)}2 Now if p' and m be both small, this expression becomes sin2 a sin2ß = μ1 (sin'm + sin'p' – 2 sin3m sin3 p' cos 20), and sin'm' sin'n' = (sin'm + sin'' - 2sin' m sin' p' cos 20); sin'p' ... sin'a sin'ß = u1 sin2m' sin2n'. If m be very small compared with p' sina sin2ßu' sin'p' = =u' sin m' sin n'. If ' be very small compared with m, sin a sinẞu' sin'm usin ̊ m' sin'n'. = = In all cases therefore, provided one of two, either p' or m be small, or if they are both small, we have |