526 SCIENCE PROGRESS 



surface which is the locus of points whose polar planes with 

 respect to a cubic surface Cg touch Ta by the plane 7 for which 

 the four-nodal cubic is the polo-cubic surface with respect to 

 Cg ; the complete configuration of the 28 tritangent plane pairs 

 of the same system may thus be deduced. 



The two inflexional tangents at a point on a surface have in 

 general three intersections with the surface at the point ; there 

 are, however, an infinite number of inflexional tangents with 

 more than three-point contact. Enumerative results in this 

 connection were obtained by Salmon for the general surface of 

 order n ) H. W. E. Jung proposes to call such tangents Salmon 

 tangents, and determines (Crelle, 152, 1922, 11-29) (0 the 

 number of places where both inflexional tangents are Salmon 

 tangents, and (2) the number of Salmon tangents which meet 

 an arbitrary line, in terms of the arithmetical genus of the 

 surface, the genera of a plane section and of the tangent cone 

 from an arbitrary point, the Zeuthen-Segre invariant, the 

 class of a plane section, and the order and class of the surface. 



O. Haupt (Crelle, 152, 1922, 6-10) defines asymptotes of 

 plane curves in a wider sense and obtains sufficient conditions 

 for asymptotic tangents and asymptotic parabolas. 



Brunn in 1889 obtained an inequahty between the areas of 

 three parallel sections of a convex body, Fj, Fg, F3, of which Fg 

 lies between the others and divides the distance between them 

 in the ratio / : (i - /), in the form 



the equality sign holding only when F^ and F, are similar and 

 similarly situated and the part of the convex body lying between 

 them consists of the frustum of a cone which joins them. Further 

 proofs have been given by Minkowski and Blaschke. K. Zindler 

 (Math. Zs., 31, 1922, 106-10) gives a proof in the elementary 

 special case of convex polyhedra, without limit considerations. 



The generalisation to three dimensions of the equiangular 

 spiral, namely the surface which cuts the rays of a sheaf at the 

 same angle, has been studied by G. Scheffers (1902), who gave 

 a general method of generating such surfaces and determined 

 their lines of curvature, and by G. Landsberg (1909), who 

 obtained the two fundamental quadratic forms. There are two 

 recent papers on the subject by R. Baldus (Math. Zs., 15, 1922, 

 147-58 ; Berichte Heidelberg, 1921, Abh. 10), who gives a series 

 of geometrical properties of the surface, and by considering also 

 imaginary surfaces obtains those which are algebraic. Equian- 

 gular spirals which are at the same time algebraic curves have 

 been considered by the same author {Archiv d. Math., 28, 1920, 

 102-1 1) ; there is, for instance, the well-known curious theorem 

 that if a rectangular hyperbola be a parabola it is also an 

 equiangular spiral. 



