Many years ago a method convenient for photographing the whole sky with the aid of a convex mirror was advocated by Dr. S. Suzuki and J. Georgi independently. In the present paper some properties bearing on the geometrical optics of the mirror are discussed with a special reference to a new mirror of distortionless projection.Here the optical axis of a camera is taken vertically downwards passing through the center of mirror, and the angles between the axis, and rays before and after the reflection at the surface of the mirror are denoted by z and θ respectively. Then z is the zenith distance. If the solid angle subtended by any of clouds can be measured directly by the area of its image on the photographic plate in a camera, the following relation must hold:orThis is the condition for solid angle projection. On the other hand when a convex spherical mirror is used instead of the above nonspherical, the corresponding relation is In (1) and (2) c and k are positive constants. If c and k are equal and large enough, then (1) and (2) express the same relation between z and θ, that is, when the photographing distance is great, a spherical mirror is the limiting one of solid ang_??_e projection. Fig. 1 explains this graphically and shows that the length of the diameter equal to the distance of the camera to the mirror meets any practical purpose of estimating the amount of clouds in solid angle. Therefore a spherical mirror may be used for solid angle projection with a good accuracy. Now, in order to get a good focus, it is desirable for us to know the location of image by the mirror and its astigmatic difference. In case of spherical mirror the location of image in quenstion can be inferend from the intersecting point of the rays which are parallel befor the reflection. In order to obtain the astigmatic difference, let us calculate the location of the primary image, which is made by the meridional rays, and the secondary by the sagittal rays (z=const.). Then the two images are distant from the center of camera lens by ρ+X and ρ+Y, respectively, where ρ is the distance from the lens to a point of mirror, then X and Y can be calculated easily to beR: radius of the spherical mirror.Of course these values are reduced to X=Y=R/2 in the limiting case when z_??_0. The form of the imaginary curved surface in which the primary image seems to lie depends on the camera distance as given in Fig. 2. From the above-mentioned the astigmatic difference is, From Fig. 1 can be seen an interesting fact that, on using a spherical mirror, several distances from a camera, that is respectively correspond to nearly (a) stereographic, (b) equidistant and (c) solid angle projections.In the second place, let us consider a distortionless projection of an apparently small object. After Robin Hill we call this a stereographic projection. If a small sphere lies afar, the ratio of its image contractions, vertical and horizontal, may be expressed in general as follows.When the above condition is satisfied, this value must be unity, that is, orThis is the relation between z and θ in a stereographic projection. When a ray is reflected at the surface of a mirror, the law of reflection must be fulfilled, that is, Eliminating z from (7) and (8), we obtain easily an analytic expression of the stereographic mirror as follows: As an example, a sectional form of this mirror is given in Fig. 3 in case of c=4. Regarding the construction and photography of stereographic mirror, it will be stated in a later report. In conclusion, the author's best thanks aree due to Dr. Suzuki for his valuable suggestion and guidance and to Dr. Okada for his everlasting encouragement.