MATHEMATICS ii 



from such transcendentals has been taken by R. Konig {Rend. 

 Circ. Mat. Palermo, 45 (1921), 284-312). He first shows how 

 such a curve arises from a class {K) of the functions and the 

 corresponding differentials ; the complementary class also comes 

 into consideration, being characteristic for the new theory. 

 Singularities on the curve are next investigated, and it is shown 

 the functions cannot be chosen arbitrarily, but must be linearly 

 independent multiples of a point set (a divisor). Part 3 

 generalises the Riemann-Roch Theorem and the Brill-Noether 

 Reciprocity Theorem to transcendental curves and investigates 

 normal curves. Part 4 shows how the algebraic case differs 

 from the more general one, and, in particular, how the different 

 ways in which the genus arises have to be generalised in different 

 ways. 



Just as with the rise of Projective Geometry attention 

 became concentrated on conies and quadrics as generalisations 

 of the circle and the sphere, so, in recent years, other generalisa- 

 tions, namely, " convex regions " and " convex bodies," have 

 come into prominence. By a " convex body " is meant a 

 point set A which is bounded and closed and such that of 

 the two segments of a straight line determined by two points 

 of /^ one always belongs wholly to A. It is remarkable that 

 from the mere fact of convexity many beautiful and important 

 results can be deduced. 



A useful summary, with references, of investigations down 

 to 19 1 6 is given in a little book by W. Blaschke of Hamburg 

 entitled Kreis und Kngel ; new results are, however, continually 

 being obtained. 



A. Speiser {Vierteljahrsschrift Zurich, 66, 1921, 28-38) 

 proves that a convex surface must in general have closed 

 geodesies. 



Several recent papers relate to convex bodies in n dimensions. 

 H. Kneser {Math. Ann., 82, 1921, 287-96) shows that progress 

 may be made even if the restriction to bounded sets is omitted. 

 J. Radon {Math. Ann.,%Zi 1921, 1 13-15) proves a theorem enun- 

 ciated by E. Helly with respect to the necessary and sufficient 

 conditions for a set of convex bodies in n dimensions to have a 

 common point. 



P. Steinhagen {Hamburg Seminar, 1, 1921, 15-26) estab- 

 lishes for the general case inequalities between the breadth b, 

 i.e. the minimum distance between two parallel tangent spaces, 

 and the radius r of the largest " sphere " which can be contained 

 in the body. He finds that, if n is odd, b < 2r'\/n, while, if n 



n+ I 

 IS even, b <2r , 



' . ^ V n + 2 



The corresponding inequalities between the diameter d, i.e. 



