New properties of the Busemann ellipsea

A combination of the Busemann ellipse, the inscribed unit circle and a circle of radius √2 about the same centre is considered. For supersonic two-dimensional potential gas flows, it is shown that the inclinations of the velocity vector in motion along an arbitrary characteristic, the characteristic itself and the characteristic of the other family have values equal to, respectively. the difference between the areas of the elliptical and circular (R = 1) sectors, the difference between the areas of the elliptical and circular (R = √2) sectors, and the area of the elliptical sector, apart from unimportant multiplicative and additive constants. The straight sides of the sectors in question are the semiminor antis of the ellipse and the radius vector of the velocity. The obvious analogy with one of Ke:pler's laws is pointed out. The existence of a point of intersection of the ellipse and the second circle illustrates a well-known result of Khristianovich concerning the points of inflexion of characteristics with a monotone velocity distribution. It is shown how the combination of the ellipse and the inscribed circle illustrates the simplification of the compatibility conditions and the Darboux equation for trans- and hypersonic flows.