pjirailel aiiiilc oorrespoiidiiig' to it is precisely 45", tlic oomplenieiit 

 becomes 45° too, jiiid therefore / parallel (o y>,, but on the other 

 side of 1)T compared to e\ I will still iiitei'sect /^^ in a finite point 

 7-*!, for as it enters the trian<>le 00J*\^ at 1), does of course 

 not contain the point P\^^, and is divergent with reference to 00^, 

 it can leave the triangle only in a finite point of />i ; but it will cnt />, in 

 an point at infinity V^j,^, being at the same time T,^. So its projectio 

 consists of the segment of the line c of T'^', over D to F, , and 

 the isolated point Z^'^,^ is eqnal to T',^ ; now too it is determiner' 

 by two of the three points D, V\, l\^^. 



The point D lies at a certain distance r from 0^ ; if we describe 

 a circle in t about O^ as centre and with r as radius, and if we 

 erect in all points of that circle the j)er[)endiculars on r, a snrface 

 appears which may be called a cylinder of revolution, of which the 

 circle Just mentioned is the gorge line; the lines / (it) lying inside 

 that cylinder have two dilferent vanishing points (with the exception 

 of 0(>i, whose projection is a single point), the lines /on the cylinder 

 have a finite and an infinite vanishing point, and the lines / ontside 

 the cylinder miss the second vanishing i)oint. 



As for the shape of the cylinder it is easy to see, that the plane 

 T* (see N". 1) is an- asymptotic plane; and t itself being evidently a 

 plane of OJ'thogonal symmetry, the plane t** normal to 00^ in the 

 point (>!**" symmetrical to (J^* with respect to r will be a second 

 asymptotic plane; so the distance of these two planes is 4r/. 



5. In P]uclidean Geometry the lines It are at the same time 

 those which are parallel to 00^, hnt in Hyperbolic Geometry this 

 is different ; here we ha\'e to regard the lines having in common with 

 OOj^ the point ["j^ lji"o ii^idei" the pictnre plane at infinity, and 

 those having with 00^ in common the point V,^ lying above t. 

 A line / of the former kind lying in the vicinity of 00^ has a 

 pictnre point D, two points 1\, P^, and a second point at infinity 

 lying inside the cone h; its first vanishing point coincides with 0^, 

 whilst the second lies on DO^ in snch a way, that 0^ lies between 

 D and that point. 



If the perpendicnlar OS let down out of to / becomes conti- 

 nually larger, the first particularity appearing here is that / becomes 

 parallel to the generatrix /;, of cone a lying in the plane 01; then 

 it is at right angles to the bisectrix of the obtuse angle formed by 

 p2 and 00^. All lines having this property form an asymptotic cone 

 of revolution') with vertex F^^, whilst t* is an asymptotic plane; 



1) H. LiEBMANN, "Nichteuklidische Geometrie", Collection Schubert XLIX, page 63. 



