lo SCIENCE PROGRESS 



example, that in four dimensions six spaces determine three 

 convex polyhedra, of which two have eight vertices and the 

 other has nine. 



On a surface in ordinary space there exists in general an 

 infinite number of conjugate sets of curves — each curve of one 

 set being met by each curve of the other in such a way that 

 the tangents at the intersection are conjugate. This is no longer 

 true in higher space, E. Bompiani {Rend. Palermo, 46, 1922, 

 91-104) defines and investigates analogous systems of curves 

 on surfaces in n dimensions. 



A. Comessatti {Rend. Palermo, 46, 1922, 1-48) applies the 

 projective differential geometry of n dimensions to classify 

 irregular algebraic surfaces in ordinary space for which the 

 geometric genus exceeds the arithmetic genus by i at least 

 and obtains the relations between the Picard simple integrals 

 of the first kind connected with such a surface. 



Darboux first investigated (i, i) point transformations 

 between two surfaces such that the configuration formed by 

 two corresponding points and the tangent planes thereat is 

 invariable when the pair of points varies ; if the surfaces are 

 not parallel they are parallel respectively either to two surfaces 

 of equal constant total curvature or to two minimal surfaces. 

 P. Tortorici {Rend. Palermo, 46, 1922, 122-45) examines in 

 detail the latter case. 



J. A. Schouten {Math. Zs., 11, 192 1, 58-88) and J. A. Schou- 

 ten and D. J, Struik {Rend. Palermo, 45, 1921, 313 ; 46, 1922, 

 165-84) investigate curvature relationships for w-dimensional 

 manifolds. 



In the Euclidean plane a circle can be defined in two ways, 

 as the locus of points equidistant from a given plint, or as the 

 curve of constant curvature. If we extend these definitions 

 to curved surfaces we get two kinds of curves ; B. Baule {Math. 

 Ann. 83, 1921, 286-310) investigates conditions for them to be 

 the same, and also examines the corresponding problem in a 

 Riemann space, of n dimensions. 



B, Gambler {C.R., 173, 1921, 763-6) discusses conformal 

 correspondences between two surfaces which conserve lines of 

 curvature and the ratio of the absolute values of the principal 

 curvatures. 



Geometry on an algebraic curve, like so many other subjects 

 of modern mathematics, was created by Riemann. He also 

 recognised that algebraic functions are a special case of functions 

 which on making a circuit round given points undergo linear 

 homogeneous substitutions, i.e. solutions of homogeneous 

 linear differential equations of Fuchs' type, the so-called Rie- 

 mann's Transcendentals. 



The step of proceeding to geometry on the curves arising 



