( 148 ) 



Mathematics. — ''The surfaces of revolution or quadratic cylinders 

 of non-Euclidean space". By Prof. J. A. Barrau. (Com- 

 nmnicated by Prof. J. Cardinaal). 



(Commuiiicaled in the meeting of April 28, 1911). 



In hyperbolic space each quadratic surface whose inter- 

 section with the absolute ((uadratic surface i2 degenerates into two 

 conies fcasuquo still farther) is surface of revolution as well as 

 cylinder, in such respect that the line of intersection of the planes 

 of the products of degeneration is cylinder axis, its reciprocal polar 

 line with respect to i2 axis of revolution. 



In one consideration the surface is generated as locus of a conic 

 revolving round one of its axes ^); in the other as locus of an invariable 

 conic, of which one of the centres describes a right line, to which its 

 plane always remains perpendicular, while its points describe plane 

 curves. It is clear, that by assuming this definition of cylinder, that 

 one as cone with vertex at infinite distance, which coincides with 

 it in Euclidean geometry, is abandoned. 



Now according to the axis of revolution being metrically real (i. e. 

 having a real part within ii) and therefore the cylinder axis ideal, or 

 the reverse, it will be more natural to regard the surface as surface 

 of revolution tlian as cylinder (classes A and B), whilst a transition 

 class C is formed by the cases in which both axes are conjugated 

 tangents to i2. 



If the surface is projectively real (i. e. metrically real or ideal) 

 then the planes of the degenerations are (projectively) either real or 

 conjugated complex; both axes are thus in any case projectivelj' real. 

 Each plane through the cylinder axis cuts the surface along a 

 conic in double contact with i^, that is along a circle after hyper- 

 bolical measure. 



If this intersection is metrically real, then it is a (finite) circle, a 

 limiting circle (or circular parabola) or a line of distance, according 

 to the surface being ranged in class A, C or B"). So we have a 

 first system of circular sections for each surface. 



But the general quadratic surface possesses four systems of 

 circular sections, namely the tangential planes to the four focal 

 cones (cones in the pencil determined by i2 and the surface). 

 Of these four systems two are absorbed in our case by the above- 



1) Gomp. Story: On non-Euclidean Properties of Conies [Amer. Journal of 

 Mathematics, vol V, p. 358). His terminology is followed here. 

 -) Some surfaces fall in more than one class. 



