Formerly the observations of twilight were executed by Fessenkoff , Smart , Hulburt . The latest one perhaps was given by Ljunghall , who introduced us the historical process and afforded many useful references for this paper. As far as the theoretical investigation is aconcerned, Hulburt computed by considering the primary scattering only, so that the error amounted up to 30% and he confessed that a rigorous calculation would be impossible for the sun's zenith-distance greater than 98°, owing to the great predominancy of the secondary scattering. Hence a satisfactory explanation of twilight has not yet been completed. After having given a paper on the scattering of the sun in the day-time , the author has now researched the twilight as an extention of that paper, assuming also the earth's atmosphere to be composed of numberless homogeneous thin layers bounded by concentric spheres, and considering the extinction of the sun's ray and further discussing strictly the secondary scattering with respect to each wave-length, and has computed the radiation of each portion of the sky dome and that falling on a horizontal plane at the earth's surface in the cases when the sun's zenith-distances are 96°, 96.°404, 97°, respectively: but as the above papers , ,  and that introduced in  are giving the observation of particular portion of the sky, and as  is concerned with the luminosity on a vertical plane, so we feel some unreliability and inconvenience in the representation of solid angle and in the method of measurement. On the contrary, as the observation by Mr. Osawa of Tokyo Astronomical Observatory  can give full reliability, and is concerned with the horizontal plane on the earth's surface by means of the unit lux, so his measurement is most convenient compared with the author's computaion. This comparison has been made and has shown a good identity. Eventually, the explanation of twilight which has not been hitherto researched is now possible up to about 100° of the sun's zenith-distance if we consider up to the secondary scattering from a practical point of view.
The essential features of the turbulent diffusion in the atmosphere exist in that the atmospheric turbulence consists of eddies, whose dimensions are very different from each other, and that the scale of eddies effective for the diffusion is related to the scale of the diffusion phenomenon. In the present paper, the author treated theoretically the turbulent diffusion from a point source by means of the relations between the statistical quantities of the turbulence and the spatial length under analysis. Using this method, it was made possible to take into consideration the circumstances that small eddies are effective for the diffusion near the point source and large eddies become effective, as the particles depart from the source. By this investigation, the so-called Lagrangian correlation coefficient was derived and the interrelation between the Eulerian and Lagrangian correlation coefficients was obtained.