THE MANUFACTURE OF INTERMEDIATE PRODUCTS FOR DYES. By JOHN CANNELL CAIN, D.Sc. (Manchester), Editor of the Journal of the Chemical Society'; Examiner in Coal-Tar Colouring Matters to the City and Guilds of London Institute; late Member of the Technical Committee, British Dyes, Ltd.; and Chief Chemist of the Dalton Works, Huddersfield. With 25 Illustrations. 8vo. 10s. net. A TEXT-BOOK OF PHYSICS FOR THE USE OF STUDENTS OF HEAT. 38. 6d. MAGNETISM AND ELECTRICITY. 4s. Nature. "The authors of this volume-one an engineer and the other a physicist-are to be congratulated on the successful way in which they have accomplished their task.. The book is most satisfactory, and should make a strong appeal to all engineering students." A CLASS-BOOK OF ORGANIC CHEMISTRY. By J. B. COHEN, Ph.D., B.Sc., F.R.S., Professor of Organic Chemistry, the University, Leeds. Globe 8vo. 4s. 6d. 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On the Scattering of Light by a Cloud of similar small Particles of any Shape and oriented at random. By Lord RAYLEIGH, O.M., F.R.S.* OR distinctness of conception the material of the par ticles may be supposed to be uniform and non-magnetic, but of dielectric capacity different from that of the surrounding medium; at the same time the results at which we shall arrive are doubtless more general. The smallness is, of course, to be understood as relative to the wave-length of the vibrations. When the particles are spherical, the problem is simple, as their orientation does not then enter t. If the incident light be polarized, there is no scattered ray in the direction of primary electric vibration, or if the incident light be unpolarized there is complete polarization of the light scattered at right angles to the direction of primary propagation. The consideration of elongated particles shows at once that a want of symmetry must usually entail a departure from the above law of polarization and may be one of the causes, though probably not the most important, of the incomplete polarization of sky-light at 90° from the sun. My son's recent experiments upon light scattered by carefully filtered gases ‡ reveal a decided deficiency of polarization in the light emitted * Communicated by the Author. + Phil. Mag. vol. xli. pp. 107, 274, 447 (1871), vol. xii. p. 81 (1881), vol. xlvii. p. 375 (1899); Scientific Papers, vol. i. pp. 87, 104, 518 vol. iv. p. 397. Roy. Soc. Proc. Feb. 28, 1918. Phil. Mag. S. 6. Vol. 35. No. 209. May 1918. 2 D perpendicularly, and seem to call for a calculation of what is to be expected from particles of arbitrary shape. As a preliminary to a more complete treatment, it may be well to take first the case of particles symmetrical about an axis, or at any rate behaving as if they were such, for the calculation is then a good deal simpler. We may also limit ourselves to finding the ratio of intensities of the two polarized components in the light scattered at right angles, the principal component being that which vibrates parallel to the primary vibrations, and the subordinate component (vanishing for spherical particles) being that in which the vibrations are perpendicular to the primary vibrations. All that we are then concerned with are certain resolving factors, and the integration over angular space required to take account of the random orientations. In virtue of the postulated symmetry, a revolution of a particle about its own axis has no effect, so that in the integration we have to deal only with the direction of this axis. It is to be observed that the system of vibrations scattered by a particle depends upon the direction of primary vibration without regard to that of primary propagation. In the case of a spherical particle the system of scattered vibrations is symmetrical with respect to this direction and the amplitude of the scattered vibration is proportional to the cosine of the angle between the primary and secondary vibrations. When we pass to unsymmetrical particles, we have first to resolve the primary vibrations in directions corresponding to certain principal axes of the disturbing particle and to introduce separate coefficients of radiation for the different axes. Each of the three component radiations is symmetrical with respect to its own axis, and follows the same law as obtains for the sphere *. In fig. 1 the various directions are represented by points on a spherical surface with centre O. Thus in the rectangular system XYZ, OZ is the direction of primary vibration, corresponding (we may suppose) to primary propagation parallel to OX. The rectangular system UVW represents in like manner the principal axes of a particle, so that UV, VW, WU are quadrants. Since symmetry of the particle round W has been postulated, there is no loss of generality in taking U upon the prolongation of ZW. As usual, we denote ZW by 0, and XZW by p. The first step is the resolving of the primary vibration Z in the directions U, V, W. We have cos ZU-sin 0, cos ZV=0, cos ZW = cos 0. . (1) * Phil. Mag. vol. xliv. p. 28 (1897); Sci. Papers, vol. iv. p. 305, The coefficients, dependent upon the character of the par- W perpendicular to both primary vibrations and primary propagation. The ray scattered in this direction will not be completely polarized, and we consider separately vibrations parallel to Z and to X. As regards the former, we have the same set of factors over again, as in (1), so that the vibration is A sin20+C cos2 0, reducing to C simply, if A-C. This is the result for a single particle whose axis is at W. What we are aiming at is the aggregate intensity due to a large number of particles with their positions and their axes distributed at random. The mean intensity is This represents the intensity of that polarized component of the scattered light along OY whose vibrations are parallel to OZ. For the vibrations parallel to OX the second set of resolving factors is cos UX, cos VX, cos WX. Now from the spherical triangle UZX, cos UX=sin (90° +0) cos &=cos e cos p. Also from the triangles VZX, WZX, cos VX=cos VZW=cos (90°+4)= −sin 4, The first set of factors remains as before. Taking both sets into account, we get for the vibration parallel to X -A sin cos cos & + C cos & sin cos 0, the square of which is (3) (C-A)2 sin2 cos2 0 cos2 p. The mean value of cos2 is. That of cos is and that of cos is, as above, so that corresponding to (2) we have for the mean intensity of the vibrations parallel to X The ratio of intensities of the two components is thus (4) Two particular cases are worthy of notice. If A can be neglected in comparison with C, (5) becomes simply onethird. On the other hand, if A is predominant, (5) reduces to one-eighth. The above expressions apply when the primary light, propagated parallel to X, is completely polarized with vibrations parallel to Z, the direction of the secondary ray being along OY. If the primary light be unpolarized, we have further to include the effect of the primary vibrations parallel to Y. The two polarized components scattered along OY, resulting therefrom, both vibrate in directions perpendicular to OY, and accordingly are both represented by (4). In the case of unpolarized primary light we have therefore to double (4) for the secondary vibrations parallel to X, and to add together (2) and (4) for the vibrations parallel to Z. The latter becomes 11⁄2 (9A2+4C2+2AC), and for the ratio of intensities of the two components. 2(C-A)2 When A=0, this ratio is one-half. (6) For a more general treatment, which shall include all forms of particle, we must introduce another angle to |