to 19. By these values of the angles, the equation in (17) is reduced - M (cos μ cos e — sin μ sin e cos у) (sin μ cos e + cos μ sin e cos ↓) μ μ -N (cos μ sine + sin μ cos e cos y) (sin μ sin e- cos μ cos e cos↓) + P sin μ cos μ sin2 =0, or -- M {sin 2μ (cos2 e - sinR e cos2 y) + cos 2μ sin 2e cos y} - N {sin 2μ (sin2 e – cos2 e cos2 †) — cos 2μ sin 2e cos } +P sin 2μ sin2 = 0, or sin 2μ {(M-N) (cos2 e-sin2 e cos2 y) + (N-P) sin2 } 20. The same substitutions reduce the value of v2 to v2 = a2 cos2 02 + b2 cos2 p2 + c2 cos2 2 = a2 (cos2 μ. cose + sinu sin2 e cos y sin 2μ sin 2e cos) +b2 (cos2 u sin2 e + sin μ cos2 e cos2 + sin 2μ sin 2e cos ) μ + c2 sin2 μ sin2; 12 Hence v2 + v22 = a2 (cos2 e + sin' e cos2 y) + b2 (sin e + cos e cos2 ) 21. Now in the equation of condition (19) since hence v22 - v2 = (a2 – b2) sin 2μ sin 2e cos y + (a2 — b2) appears from the above equation for cot 2u (a), that ... sin 2μ= 0. This value therefore does not make the expression vanish, as would appear at first sight; it can vanish only, when sin 2€ = 0; and the vibration is either in the plane of xz or of yz. And from equation (a), if e = 0, a2 − b2 + (b2 — c2) sin2 ↓ = 0, and must be intermediate to a and c. Both these cannot be true. for the same medium: let the latter only be true, then the transmission is in the plane of xx, and there are two directions, one on each side of the axis of x, for which the velocities of transmission of both vibrations are the same, which directions are the optic axes*. We will call m the angle made by this optic axis with the axis of, so that b2 a2 a2 b2 c2. (a3 — b2) (cos3 e — sin2 e cos3 \) + (b2 — c2) sin2 √ (a2 = b2) sin 2e cos cos2 e - sin2 e cos2 + cot2 m sin' y sin 2 e cos y The optic axis here is not the same as that which Fresnel calls by the same name. This is determined by the direction of the wave, his by that of the ray. As I shall have to compare them, I will use the term radial axis instead of optic axis when speaking of the latter. Now if O and R in the figure (18) be the two optic axes, it is evident that therefore the plane which defines one vibration bisects the angle between the planes passing through the normal to the front of the wave and these two optic axes. The plane which defines the other is manifestly at right angles to this. 23. We saw in (21) that the expression for the difference of the squares of the velocities is 24. From M. Fresnel's construction it appears that the sum of the squares of the velocities of the two waves perpendicular to their front, and travelling in the same line (we are speaking of vibrations not of the motion of the rays conveyed by them) is (Ency. Met. Light, p. 544.) a2 + b2 + m2 (b2 + c2) + n2 (a2 + c2) 1 + m2 + n2 is the equation to the plane in which the vibra where x = mx + ny |
