449 along x, the former being applicable when the vibration is parallel to y, and the latter when it is parallel to x. The directions of vibration are determined by (5) and by the consideration that (I, m, n), (A, /n, v}, and ( 2 , &V> c2 ") lie in a plane, or (as we may put it) are all perpendicular to one direction Thus (8). The determinant expressing the result of the elimination of/: g: h may be put into the form which with (5) suffices to determine (A, /j., v} as a function of (I, m, n}. The fact that the system of equations (5), (8) is symmetrical as between (A, /j., v) and (/, g, h) proves that the two directions of vibration corresponding to a given (I, m, n) are perpendicular to one another. The direct investigation of the wave-surface from (4) and (7) was first effected by Ampere, but his analytical process was very labori ous. Fresnel had indeed been forced to content himself with an indirect method of verification. But in the following investigation Smith s of A. Smith 1 the eliminations are effected with comparatively little /estiga- trouble. in. In addition to (4) and (7), we know that -700,01 /I AN l- + m- + n- = l (10). To find the equation to the envelope, we have to differentiate these equations, making I, m, n, v vary. Eliminating the differentials by the method of multipliers, we obtain the following: 2 ) (11), ^ (12), z = An + En/(v 2 -c 2 ) (13); and ( T 2 ni z n 2 i 1T> I ^ //(//(// - . x = Bu< r - 5 oTo + r^ T2V 2 + r~2 2 ( ( 14 )- ( (v* - a~Y (v - b^y (ir - c J ) J ) The equations (11), (12), (13) multiplied by I, m, n respectively and added, give t>-A . (15). The same equations, squared and added, give x 2 + y 2 + z 2 = A 2 + B/v . If we put r 2 for x 2 + y 2 + z 2 , and for A the value just found, we obtain If these values of A and B be substituted in (11), r 2 -a 2 X=lV < 1+
- O
(17). If we substitute this value of I, and the corresponding values of m, n in (4), we get (v 2 - a~) x 2 (ir -b~)y 2 (v 2 - c 2 ) 2 2 _ 2 _ r 2 a 2 r 2 b 2 r 2 c 2 r 2 r* whence esnel s ive- rface. 2 +J^ + - T n 7 ) I r 2 - b 2 r" - <? as the equation of the wave-surface. By (6) equation (11) may be written (18), from which and the corresponding equations we see that the direc tion (x, y, z) lies in the same plane as (I, m, n) and (A, /u, v). Hence in any tangent plane of the wave-surface the direction of vibration is that of the line joining the foot of the perpendicular and the point of contact (a;, y, z). The equation (18) leads to another geometrical definition of Fresnel s wave-surface. If through the centre of the ellipsoid reciprocal to the ellipsoid of elasticity (3), viz., X 2 /a 2 + y 2 /b 2 + Z 2 /C 2 =l ..... (19), a plane be drawn, and on the normal to this plane two lengths be marked off proportional to the axes of the elliptic section deter mined by the plane, the locus of the points thus obtained, the apsidal surface of (19), is the wave-surface (18). Fully developed in integral powers of the coordinates, (18) takes the form (a; 2 + 1/ 2 + s 2 )(a 2 x 2 + 6 V + c 2 z 2 ) - a 2 (b 2 + c 2 )x 2 l> 2 c 2 = Q . . . (20). incipal The section of (20) by the coordinate plane ?y Jtions. 2222 2222 . . . (21), 1 Camb. Trans., vi., 1835. representing a circle and an ellipse (fig. 24). That the sections by each of the principal planes would be a circle and an ellipse might have been foreseen independently of a general solution of the en velope problem. The forms of the sections prescribed in (21) and the two similar equations are suffi cient to determine the character of the wave-surface, if we assume that it is of the fourth degree, and involves only the even powers of the coordi nates. It was somewhat in this way that the equation was first obtained by Fresnel. If two of the principal velocities, e.g., a and b, are equal, (20) becomes (x 2 + y 2 + z 2 a 2 )(a 2 x 2 + a 2 y 2 + c 2 z 2 - a 2 c 2 ) = . (22), so that the wave-surface degenerates into the Huygenian sphere and ellip soid of revolution appropriate to a Fig. 24. uniaxal crystal. The two sheets touch one another at the points a: = 0, y = 0, 2=. If oa, as in Iceland spar, the ellipsoid is external to the sphere. On the other hand, if c<a, as in quartz, the ellipsoid is internal. We have seen that when the wave-front is parallel to the circular sections of (3), the two wave velocities coincide. Thus in (7), if a 2 , b 2 , c 2 be in descending order of magnitude, we have m = 0, v = b ; so that ^ . n 1 (23^ o jo 79 o o .) V* 11 */ 1 a 2 -tr b 2 - c 2 a 2 - c 2 In general, if 6, be the angles which the normal to the actual wave-front makes with the optic axes, it may be proved that the difference of the squares of the two roots of (7) is given by V 2 2 ~v 1 2 =(a 2 ~c 2 }sin8siue .... (24). In a uniaxal crystal the optic axes coincide with the axis of symmetry, and there is no distinction between 6 and 6. Since waves in a biaxal crystal propagated along either optic axis have but one velocity, it follows that tangent planes to the wave- surface, perpendicular to these directions, touch both sheets of the surface. It may be proved further that each plane touches the surface not merely at two, but at an infinite number of points, which lie upon a circle. The directions of the optic axes, and the angle included between them, are found frequently to vary with the colour of the light. Such a variation is to be expected, in view of dispersion, which renders a 2 , b 2 , c 2 functions of the wave-length. A knowledge of the form of the wave-surface determines in all cases the law of refraction according to the construction of Huygens. We will suppose for simplicity that the first medium is air, and that the surface of separation between the media is plane. The incident wave-front at any moment of time cuts the surface of separation in a straight line. On this line take any point, and with it as centre construct the wave-surface in the second medium corresponding to a certain interval of time. At the end of this interval the trace of the incident wave-front upon the surface will have advanced to a new position, parallel to the former. Planes drawn through this line so as to touch the wave- surface give the positions of the refracted wave-fronts. None other could satisfy the two conditions (l)that the refracted wave- front should move within the crystal with the normal velocity suitable to its direction, and (2) that the traces of the incident and refracted waves upon the surface of separation should move together. The normal to a refracted wave lies necessarily in the plane of incidence, but the refracted ray, coinciding with the radius vector of the wave-surface, in general deviates from it. In most cases it is sufficient to attend to the wave normal. As in total reflexion by simply refracting media, it may happen that no tangent planes can be drawn to satisfy the prescribed conditions, or that but one such can be drawn. When the crystal is uniaxal, one wave is refracted according to the ordinary law of Snell. The accuracy of both the sphere and the ellipsoid of the Huygenian construction has been fully verified by modern observations. 2 The simplest case of uniaxal refraction is when the axis of the crystal is perpendicular to the plane of incidence, with respect to which every thing then becomes symmetrical. The section of the wave- surface with which we have to deal reduces to two con centric circles ; so that both waves are refracted according to the ordinary law, though of course with different indices. In biaxal crystals one wave follows the ordinary law of refrac tion, if the plane of incidence coincide with a principal plane of the crystal. This consequence of his theory was verified by Fresnel himself, and subsequently by Rudberg and others. But the most remarkable phenomena of biaxal refraction are undoubtedly Uniaxal crystals. Optic axes. Huygens construc tion Total re flexion. 2 Stokes, Proc. Roy. Soc., vol. xx. p. 443, 1872; Glazebrook, Phil. Trans., 18SO, p. 421 ; Hustings, Amer. Jour., Jan. 1S8S.
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