( 39Ó ) 



In the following a few observations will be given on these two 

 questions. 



Let lis suppose an arbitrary plane e through the line OOj, standing 

 therefore at right angles to t ; in that plane we can draw through 

 two straight lines ^^i,^).^ parallel to the line of intersection e of e 

 and T passing through Oj, therefore also parallel to t itself. The 

 angles formed by i»^ and p., with 0()^ are equal; they are both 

 acute, and their amount is a function of the distance 00^ =z d. 

 LoBATSCHEPSKY has Called each of these two angles \he parallel cüigle'^) 

 belonging to the distance d, and has indicated them by /Z^^, ; if d is 

 given, the parallel angle is found out of the relation 



in which for the number e the basis of the natural logarithmic 

 system may be taken, if only the unity of length l)y which 00^ is 

 measured be taken accordingly ^). As far as the range of values 

 of /7(d) is concerned, I only observe that the parallel angle = 7^ jt 

 for d = 0, decreasing and tending to if d increases and tends 

 to GC. 



If the plane s rotates round 00^, then p^ and p-i ^^ill describe a 

 cone of revolution round OO^ as axis; this cone is the locus of all 

 straight lines through ('> parallel to t, and distinguishes itself in many 

 respects, in form and properties, from the cone of revolution of 

 Euclidian Geometry; the plane r is an asymptotic plane, to which 

 its surface tends unlimited, and from the symmetry with respect to 

 follows easily that another plane t* like this exists, also placed 

 perpendicularly on 00^, but in the point O^* situated symmetrically 

 to Oj with respect to Ü. So the cone is entirely included between the two 

 planes t and t*, and these two planes having not a single point in 

 common (neither at Unite nor at intinite distance), are dwergent ; 

 however, they possess the common perpendicular 0^0^^, and their 

 shortest distance is 2d. The cone discussed here will be called for 

 convenience, sake the parallel cone x belonging to the point O. 



2. The parallel cone divides the space into three separate parts; 

 let us call those two parts, inside which is the axis OOj, the interior 

 of the cone, the remaining part the exterior; it is then easy to see 

 that the points of space behave dilferently with respect to their 

 projectability according to their lyiiig inside or outside the cone; 

 for a point P inside the cone the projecting ray OF forms with 



1) F. Engel: „N. I. Lobaïschefsky. Zwei geomelriscUe Abhandkingen". Leipzig, 

 Teubner, 1899, p. 167. 

 3) F. Engel, 1. c. p. 214. 



