( IM ) 



Two-sheeted surface, one inner domain, Iwo outer domains; cylinder 

 with direcli'ix degenerated into two right lines. 



In the pencil foi-med with JTi is a second cone of the same type. 



20. Hyperbolic cylinder first kind. 



Two-sheeted, non rectilinear surface. One inner domain, two outer 

 domains. Possesses a second system of distance lines in tangential 

 planes to one of the cones 19. 



21. Hyperbolic cylinder second kind. 



Two-sheeted rectilinear surface. Is generated by revolution of a 

 real right line about an ideal axis. 



XII. Planes of degeneration conjugate imaginary, 

 but not touching ^ (compare VII). 



22. Elliptic cylinder. 



One-sheeted, non rectilinear surface. In the pencil with £1 is an 

 ideal cone, whose tangential planes cut the surface along a system 

 of finite circles. These planes make equal angles (on either side) 

 with the plane of the orbit of the great axis of the directing ellipse.^) 



C. Transition Class. 

 Axes touch i2, first system of limiting circles. 



The surfaces 17 and 18. 



XIII. Planes of degeneration real. 



23. Cone of revolution icith ideal vertex, axis touching i2. 



24. Limiting hyperbolic paraboloid of revolution, first kind. 

 One-sheeted, rectilinear surface lying between 23 and the planes 



of degeneration. Is generated by revolution of a real right line about 

 an axis touching 52. 



25. Limiting hyperbolic paraboloid of revolution second kind. 

 One-sheeted, non rectilinear surface outside 23. Has a system of 



distance lines in tangential planes to 23. 



26. Limiting hyperbolic paraboloid of revolution third kind. 

 Two-sheeted, non rectilinear surface inside 23, yet outside the planes 



of degeneration. One inner domain, two outer domains. 



XIV. One plane of degen erati on real, one touching ii. 



27. Limiting semi-circular paraboloid of revolution. 

 One-sheeted, non rectilinear surface. 



XV. Planes of degeneration conjugate i m a g i n a ry. 



28. Limiting elliptic paraboloid of revolution. 

 One-sheeted, non rectilinear surface. 



1) In Euclidean geometry this quadratic system degenerates into two linear 

 systems (pencils of parallel planes). 



