( 152 ) 



In elliptic space the same considerations hold, but i2 is 

 imaginary, in consequence of which tlie number of cases remains 

 more limited. 



Ill the first place space is now finite — so each surface is closed. 

 Tlieii both axes are always real, so that each surface of revolution 

 is as naturally a cylinder. 



Finally (here exists only one real type of conic by means of which 

 the surface can be generated : the ellipse. It has three centres (of which 

 one is in the inner domain) and three axes (of which one is in the 

 outer domain). There is also but one type of circle. 



The surfaces possess — if nothing further is said — only the 

 first system of circular sections. 



We can now distinguish : 



I. P 1 a n e s of degeneration real, differing. 



1. Flattened ellipsoid of recolution. 



Non rectilinear. Is generated by revolution of the ellipse about that 

 axis cutting it, which measured in the inner domain is the shortest. 



II. Planes of degeneration real, coinciding. 



2. Spliere. 



Non rectilinear. Locus of points at fixed distance of given point, 

 likewise of given plane; oo- systems of circular sections. 



III. Planes of degeneration imaginary, not touching i2. 



3. Cone of revolution. 



4. Elongated ellipsoid of revolution. 



Non rectilinear, is generated by revolution of the ellipse about the 

 longest axis in the inner domain. 



5. Elliptic cylinder. 



Rectilinear surface, is generated by revolution of the ellipse about 

 the outer axis ; likewise by revolution of a right line about an other 

 right line, to which it is not a Clifford parallel. The tangential 

 planes to the cone (of type 3) belonging to the pencil formed with 

 i2 form a (quadratic) second system of circular sections. 



The surface has a gorge and an equator, lying in mutually 

 perpendicular surfaces. 



IV. Planes of degeneration imaginary, touching fi. 



6. Circular cylinder. 



7. Rectilinear surface. Both axes are equivalent, the surface is 

 generated in two ways by revolution of a circle around the outer 

 axis, likewise in two ways by revolution of a line about an axis 

 to which it is a Clifford parallel. It possesses two systems of circles 

 (in pencils of planes through both axes). The circles of each system 



