( 149 ) 



mentioned first s^^stem, so in general two are still to be expected, 

 which can however be ideal or imaginary (in which case we shall 

 not describe them) or they can coincide. 



We shall now give an enumeration of the possible types of these 

 surfaces, including the cones of revolution, excluding however the 

 purely ideal forms/ j 



A. Surfaces of revoi-ution Proper. 

 Cylinder axis ideal (or indefinite), first system of (finite) circles. 



I. Both planes of degeneration in e t r i (; a 1 1 y real. 



1. Cone of revolution ivith real vertex and real axis. 



2. Cone of revolution loith ideal vertex and real accis. 



Has a gorge-circle with centre in the vertex of 1 and a system 

 of distance lines in tangential planes to cone 1 . 



3. Hyperholoid of revolution first kind. 



Two-sheeted, non rectilinear") surface falling between 1 and the 

 planes of degeneration. Divides space (inside il) into one outer 

 domain (in the ordinary projective sense) and two inner domains. 



4. Hyperboloid of revolution second kind. 



Two-sheeted, non rectilinear surface falling outside the planes of 

 degeneration. One inner domain, tw^o outer domains. 



5. Hyperboloid of revolution third kind. 



One-sheeted, rectilinear surface between 1 and 2. Is generated by 

 rexolution of a real right line around a real axis. Has a gorge-circle 

 and a system of distance lines in tangential planes to 1. 



6. Hyperholoid of revolution fourth kind. 



One-sheeted, non rectilinear surface, outside 2. Has a gorge-circle 

 and two systems of distance lines, resp. in tangential planes to 

 1 and 2. 



II. n ^ plane of degeneration metrically real 

 (Z^i) one touching i2 (e. g. in /-*). 



7. Limiting cone of revolution (vertex in i2, real axis). 



8. Hyperbolic paraboloid of revolation first kind. 

 One-sheeted, non rectilinear surface between 7 and D^. Right lines 



out of P (within 7) intersect first the surface, then D^. 



9. Hyperbolic paraboloid of revolation second kind. 

 One-sheeted, non rectilhiear surface outside öi. Right lines out of /^ 



(inside 7) intersect first D^ then the surface. 



1) Of course already well known forms are again included in the place where 

 they fit in this classificalion. 



2) i. c. without real right lines. 



