Lord Rayleigh's Theory.—A more detailed investigation was conducted by the third Lord Rayleigh (Phil. Mag. XLI., 107, 275) based on an elastic-solid theory of light. This enquiry showed that both the intensity and the polarization could be satisfactorily accounted for on such a theory, if (and only if) the vibrations are perpendicular to the plane of polarization and the difference between the substance of the particles and that of the surrounding medium is one of density only.
Later (Phil. Mag. XII. 81–ioi, 1881) Rayleigh examined the question from the point of view of the electromagnetic theory in which the particles are treated as dielectric spheres. Max well's equations can be applied exactly in the case of vanishingly small spheres. The azimuth of the electric displacement travelling in any direction in the scattered wave is at right angles to that direction and is in the plane containing the scattered ray and the azimuth of the incident displacement; further the intensity is proportional to the square of the sine of the angle between these two lines. It follows that when this angle is zero the scattered light in the given direction is zero. This occurs in the direction at right angles to the incident ray. Thus, in this case, if un polarized light is incident the light scattered at right angles is completely polarized.
According to either theory, sunlight in penetrating through the earth's atmosphere should fall off according to an exponential law for each colour.
In order to test the theory Rayleigh compared the blue light of the sky taken from near the zenith with sunlight diffused through white paper. For different wave-lengths the ratios cal culated from the formula are given together with observed values for comparison:— It appears that the sky light when compared with that diffused through white paper was bluer than that required by theory; but this may possibly arise from yellowness of the paper or from the yellowness of the sunlight when it reaches us compared with its colour at higher levels.
A much more important calculation has reference to the size and number of the particles concerned in the production of the blue of the sky. Since the light scattered by each particle is proportional to the square of its volume the total amount scat tered per unit volume depends not only upon the quantity of mat ter therein but upon its fineness of division also. Assuming that the molecules of the air are the effective scatterers Rayleigh in 1899 calculated (taking Maxwell's value 19X for the num ber of molecules per unit volume of a gas at standard pres sure and temperature) that sunlight should diminish to of its value in passing through a distance of 83 kilometres in normal air. This value might agree roughly with the visibility of Mount
Everest from Darjeeling but Rayleigh considered that it implied too high a visibility since there is certainly suspended matter to be reckoned with as well. Small particles of saline or other mat ter (including organic germs) must play a part and to them may be attributed much of the bluish haze by which the moderately distant landscape is often suffused.
American Investigations.—In recent years considerable at tention has been paid to this question in America where advan tage could be taken of the remarkable clearness and dryness of the air above Mount Wilson in California. F. E. Fowle in these investigations has obtained values of the transparency coefficients for zenith observations for 3o different wave-lengths between 0.34 /2 and 2.24 ,u (i2= one millionth of a metre). The logarithms of the observed coefficients were plotted as ordinates against the corresponding quantities of precipitable atmospheric moisture as abscissae. The curves which were very nearly straight were extrapolated to zero moisture so as to obtain the transparencies for perfectly dry air. From these values for dry air the number of molecules per c.c. was calculated by means of Rayleigh's for mula. The value obtained was while the best value obtained by Millikan by other methods is 2.705X The con clusion drawn from this result is that for the clear air above Mt. Wilson the scattering is almost entirely due to the molecules of air themselves. (Fowle, Astroph. J. 1914.) Tyndall's Residual Blue.—The experiments of Tyndall upon precipitated clouds have been mentioned. When the precipitated particles are very fine, the light dispersed in a perpendicular direc tion is sky-blue and fully polarized. At a further stage of their growth the particles disperse in the perpendicular direction a light which is no longer fully polarized. When quenched as far as possi ble by rotation of a nicol prism, it exhibits a residue of a more in tense blue colour; and further it is found that the direction of the most nearly complete polarization becomes inclined to the direction of the primary rays.