Hence But if plane at P, cos ido cos &1 801 and therefore But @= be the perpendicular let fall from O on the tangent p cos v. radii of curvature at P is (rt — s2) v2 + (&c.) v + (1 + p2 + q2)2 = 0 ; and therefore if v1, v2 denote the principal radii of curvature, (1 + p2 + q2)2 rt - s2 which proves the proposition enunciated. In the particular case of an ellipsoid of revolution of which w is the axial and m the equatorial semi-axis, compared with a sphere of radius unity, both having their centres at O', one of the principal radii of curvature is the normal of the elliptic section, which by the properties of the ellipse is equal to m', m' denoting M'3 m n the semi-conjugate diameter; and the other is the radius of curvature of the elliptic section, or Also is the perpendicular let fall from the centre on the tangent line of the section. Hence from (5) or (6) mn since am' =mn. since the R, and of equation (3). This agrees with Mr Stewart's result (p. 197), ON THE INTENSITY OF THE LIGHT REFLECTED FROM OR TRANSMITTED THROUGH A PILE OF PLATES. [From the Proceedings of the Royal Society, January 23, 1862.] THE frequent employment of a pile of plates in experiments relating to polarization suggests, as a mathematical problem of some interest, the determination of the mode in which the intensity of the reflected light, and the intensity and degree of polarization of the transmitted light, are related to the number of the plates, and, in case they be not perfectly transparent, to their defect of transparency. The plates are supposed to be bounded by parallel surfaces, and to be placed parallel to one another. They will also be supposed to be formed of the same material, and to be of equal thickness, except in the case of perfect transparency, in which case the thickness does not come into account. The plates themselves and the interposed plates of air will be supposed, as is usually the case, to be sufficiently thick to prevent the occurrence of the colours of thin plates, so that we shall have to deal with intensities only. On account of the different proportions in which light is reflected at a single surface according as the light is polarized in or perpendicularly to the plane of incidence, we must take account separately of light polarized in these two ways. Also, since the rate at which light is absorbed varies with its refrangibility, we must take account separately of the different constituents of white light. If, however, the plates be perfectly transparent, we may treat white light as a whole, neglecting as insignificant the chromatic variations of reflecting power. Let p be the fraction of the incident light reflected at the first surface of a plate. Then S. IV. 10 1-p may be taken as the intensity of the transmitted light*. Also, since we know that light is reflected in the same proportion externally and internally at the two surfaces of a plate bounded by parallel surfaces, the same expressions p and 1-p will serve to denote the fractions reflected and transmitted at the second surface. We may calculate p in accordance with Fresnel's formulæ from the expressions according as the light is polarized in or perpendicularly to the plane of incidence. In the case of perfect transparency, we may in imagination. make abstraction of the substance of the plates, and state the problem as follows:-There are 2m parallel surfaces (m being the number of plates) on which light is incident, and at each of which a given fraction p of the light incident upon it is reflected, the remainder being transmitted; it is required to determine the intensity of the light reflected from or transmitted through the system, taking account of the reflexions, infinite in number, which can occur in all possible ways. This problem, the solution of which is of a simpler form than that of the general case of imperfect transparency, might be solved by a particular method. As, however, the solution is comprised in that of the problem which arises when the light is supposed to be partially absorbed, I shall at once pass on to the latter. In consequence of absorption, let the intensity of light traversing a plate be reduced in the proportion of 1 to 1 - qdæ in passing over the elementary distance do within the plate. Let T be the thickness of a plate, and therefore Tseci the length of the path of the light within it. Then, putting for shortness * In order that the intensity may be measured in this simple way, which saves rouble in the problem before us, we must define the intensity of the light transmitted across the first surface to mean what would be the intensity if the light were to emerge again into air across the second surface without suffering loss by absorption, or by reflexion at that surface. 1 to g will be the proportion in which the intensity is reduced by absorption in a single transit. The light reflected by a plate will be made up of that which is reflected at the first surface, and that which suffers 1, 3, 5, etc. internal reflexions. If the intensity of the incident light be taken as unity, the intensities of these various portions will be p, (1 − p)2 pg2, (1 − p)2p3g4, etc. and if r be the intensity of the reflected light, we have, by summing a geometric series, Similarly, if t be the intensity of the transmitted light, which is in general less than 1, but becomes equal to 1 in the limiting case of perfect transparency, in which case g 1. The values of μ, i, and q in any case being supposed known, the formulæ (1), (2), (3), (4), (5) determine r and t, which may now therefore be supposed known. The problem therefore is reduced to the following:-There are m parallel plates of which each reflects and transmits given fractions r, t of the light incident upon it: light of intensity unity being incident on the system, it is required to find the intensities of the reflected and refracted light. Let these be denoted by (m), (m). Consider a system of m+n plates, and imagine these grouped into two systems, of m and n plates respectively. The incident light being represented by unity, the light (m) will be reflected from the first group, and (m) will be transmitted. Of the latter the fraction (n) will be transmitted by the second group, and (n) reflected. Of the latter the fraction (m) will be transmitted by the first group, and (m) reflected, and so on. Hence we get for the light reflected by the whole system, $ (m) + (¥m)2 $ (n) + (¥m)2 † (m) (†n)2 + ....., and for the light transmitted, † (m)† (n) +¥ (m) † (n) $ (m) ¥ (n) + ¥ (m) ($n)2 (μm)2 ¥(n)+ which gives, by summing the two geometric series, $ (m + n) {1 − 4 (m) $ (n)} = $ (m) + $ (n) {(&m)2 − ($m)3}; and the first member of this equation being symmetrical with respect to m and n, we get, by interchanging m and n and equating the results, or $(m)+4(n) {(†m)2 — ($m)2} = $ (n) + $ (m) {(↓n)2 — ($n)2} ; which is therefore constant. ence by 2 cos a, we have Squaring (7), and eliminating the function by means of (8), we find {1 − † (m) $ (n)}2 {1 − 2 cos a . 4 (m + n) + [$ (m+n)]2} = {1 − 2 cos a . † (m) + ($m)2} {1 − 2 cos a . † (n) + ($n)2}.....(9). From the nature of the problem, m and n are positive integers, and it is only in that case that the functions p,, as hitherto defined, have any meaning. We may, however, contemplate functions, of a continuously changing variable, which are defined by the equations (6) and (7); and it is evident that if we can find such functions, they will in the particular case of a positive integral value of the variable be the functions which we are seeking. In order that equations (6), (7) may hold good for a value zero of one of the variables, suppose n, we must have $(0) = 0, † (0) = 1. The former of these equations reduces (9) for n=0 to an identical equation. Differentiating (9) with respect to n, and after differentiation putting n = 0, we find p' (0) (m) {1 – 2 cosa. 4 (m) + (pm)2) + cos a. p' (m) — 4 (m) p′ (m) = cos a. p' (0) {1 − 2 cos a . † (m) +($m)3}, |