CRYSTALLINE REFLECTION AND REFRACTION. 275 For the second medium u1 = A1 (cos X1 cos AP1 - sin x cos r1) - B1 cos R) v1 = A1 (cos X1 cos BP, + sin x, sin r1) + B, sin R...(26). 1 = Since du/dy du/dy when x = 0, and du/dz = du1/dz = 0 (23) and (24) give (A cos BP - A' cos BP') кl + Вêу sin I = A1 (cos X1 cos BP1 + sin x1 sin r1)î11 + B ̧«у, sin R.................... (27), (A cos CP - A' cos CP′) îl = à ̧ ̧l cos x1 cos CP1....... ..(28). Since m +n and m' + n are ultimately zero, and dv/dy = dv1/dy, both sides of (22) ultimately become identically equal, and this equation need not therefore be considered. Now, if A, A, be the wave-lengths of the waves S, S1; U, U1 their velocities of propagation, 1 = 2π Δι A1 = 2π sin i = λ = &c., and therefore, since U, U, are ultimately zero, A, A, are also ultimately zero; whence I = 0, R=0, and therefore, y are ultimately infinite. Also, Writing out the equation u = u, in full, multiplying by κm, and subtracting from (27), we obtain (A cos BP - A' cos BP') îl — (A cos AP + A' cos AP′) Âm A1 (cos X1 cos AP1- sin x, sin r1) m.............(29). Xi Χι From the preceding investigation we see, that B and B are not zero, but finite, and therefore the existence of the waves S, S1 cannot be entirely ignored; but, since I = R = 0 the terms involving B, B, disappear from the equation v = v', which gives A cos BP + A' cos BP' A, (cos X, cos BP, + sin x, sin r1)... (30), and the equation ww' gives = = A cos CP+A' cos CP' A, cos x cos CP......(31). Equations (28), (29), (30), and (31) contain the complete solution of the problem. with similar expressions for cos AP1, &c.; also, cos CP = cos 0, cos AP' = sin i sin e', cos BP' : cos i sin e', cos CP' = cos 0', whence (28), (29), (30) and (31) become (A cos - A' cos 0') cot i A cos + A' cos = A1 cot r1 cos x1 cos 1 A, cosec r, cos x1 sin 1 = 1 A1 (cos r1 cos X, sin 01 + sin r, sin x1) 1 = A, cos X1 cos 01 .(32), in which equations we are to recollect, that we are to add to the right-hand sides terms in A, similar to those involving A1. The preceding equations may also be obtained by a process, which does not involve the introduction of the dilatational waves. Since the continuity of u, v, w involves the continuity of their differential coefficients with respect to y and z, the first of (21) together with (23) and (24) involve the continuity of the rotations and ; also, since m = m' = — n, both sides of (22) are identically equal, and therefore this equation disappears; we are thus left with the last two of (21). The surface conditions are therefore which furnish four equations to determine the four unknown quantities. Equations (32) determine the amplitudes of the reflected and refracted waves, but according to § 10, the intensity is to be measured by the mean energy per unit of volume. Accordingly by § 269, if I, I, I, I, denote the square roots of the intensities of the four waves 2 (I cos 0+ I' cos 0') sin i = 1, cos 0, sin r1 + I, cos 0, sin r2 (I cos -I' cos 0') cos i = I, cos 0, cos r1 + I, cos 2 cos r2 I sin 0 + I' sin e' I, sin 0, + I, sin 02 (I sin - I' sin ') sin 2i = 1, (sin 0, sin 2r,+2 sin2 r, tan X1) + I2 (sin 0, sin 2r, + 2 sin2 r, tan X2) ...(33). 271. We shall hereafter show, that these equations are exactly the same as those furnished by the electromagnetic theory, and we shall postpone the complete discussion of them, until we deal with that theory; but it will be desirable to consider the results to which they lead, when both media are isotropic. 1st. Let the light be polarized in the plane of incidence; then 00' 01 = 0; X1 = X2 = 0, and I,= 0, whence = which are the same formulæ as those obtained by Fresnel. 2nd. Let the light be polarized perpendicularly to the plane of incidence; then 00' 0,=, and = which are Fresnel's formulæ for light polarized perpendicularly to the plane of incidence. See §§ 172 and 175. 272. The principal results of this theory are, (i) that it leads to a wave-surface, which is approximately though not accurately Fresnel's wave-surface,, unless k is absolutely and not approximately equal to -n; (ii) that although the direction of vibration within the crystal is not the same as in Fresnel's theory (being perpendicular to the ray instead of to the wave-normal), yet it makes the vibrations of polarized light on emerging from the crystal, perpendicular to the plane of polarization; (iii) the equations, which determine the intensities in the case of crystalline reflection and refraction are, as we shall hereafter see, identical with those which are furnished by the electromagnetic theory; and when both media are isotropic, the results agree with those obtained by Fresnel. Also, as soon as the assumptions have been made, that k is equal, or nearly so ton, and that double refraction arises from the circumstance, that crystalline media behave as if they were æolotropic as regards density; results which can be proved to be very approximately true, are capable of being deduced without the aid of any of those additional assumptions, which in many cases are indispensable in order to obtain a particular analytical result. Theory of Rotatory Polarization. 273. When bodily forces act upon the ether, the equations of motion will be of the form Now the photogyric properties of quartz and turpentine must be due to the peculiar molecular structure of such substances; and we may endeavour to construct a theory of rotatory polarization, by supposing that the effect of the mutual reaction of ether and matter modifies the motion of ethereal waves in a peculiar manner, which may be represented by the introduction of certain bodily forces. The mathematical form of these forces is a question of speculation, but we shall now, following MacCullagh', show that 1 Trans. Roy. Irish Acad. Vol. xv. p. 461. THEORY OF ROTATORY POLARIZATION. 279 rotatory polarization may be accounted for by supposing that these forces are of the form 274. For an isotropic medium such as syrup or turpentine, Px=P1 = Pz; P1 = P = Ps. If therefore the axis of z be taken as the direction of propagation, u and v will be functions of z alone; whence (34) combined with (35) reduce to where L and M may be complex constants, 7 is the period, and V the velocity of propagation. Substituting in (36), and putting U2=n/p, P3/p=p, we obtain Now the rotatory effect, which depends on p, is very small, whence in the terms involving p, U may be written for V; we accordingly obtain |