F.mch: involutoric collineations in plane and in space. 213 



an invariant ray with those two points as double-points. Every 

 point of the ray is transformed into a point of the same ray, such 

 that the rross-ra/io oi the two corresponding points and the double- 

 points is harmonic. Passing a plane through g it will intersect 1 

 in an invariant point and conversely every plain through 1 intersects 

 g in an invariant point. Planes through g and 1 are therefore 

 invariant planes. Taking such a plane J through either g or 1, it 

 will intersect the other in an invariant point C. Every ray through 

 C in J is invariant; the involution in the plain J is therefore a 

 perspective involution with C as a center and with that one of the 

 lines g and 1 through which the plane passes as an axis. 



If any ray h in space is given, the corresponding ray h, in the 

 involution is found by the following construction: Through g 

 pass any two planes J and K intersecting 1 in C and D and h in 

 A and B. In the involutions of the planes J and K determines 

 the corresponding points to A and B, respectively, which in the 

 figure are designated by Aj and B,. The ray through these points 

 is the corresponding ray to h. From this is seen immediately that, 

 in the involution of tlic scco)ui kind in space, to anv rav in space 

 corresponds a ray si/c/i tliaf tliey are on f/ie same ruled lixperl>oloid with 

 the rays of iiroariant points. 



Since the rays g and 1 and a third line 'determine a ruled 

 hyperboloid it is also seen that /// tliis in-eolution the generatrices of 

 any hyperholoid through the rays of invariant points are transformed 

 into generatrices of the same liyperludoid ; <>/■ in otiier loords, the invo- 

 lution leaves such hvperholoids invariant. 



As 00 - rays pass through a point in space there are also 00 - hy- 

 perboloids of this kind passing through a point in space. The 

 points of a line determine all the hyperboloids through g and 1, 

 their number is therefore 00 ' "^'00 -=00 '^ . Thus we may say: 



The second kind of involi/tions in space leaves the poi/its of tivo ravs 

 and the complex op hyperboloids thi-ough these ravs i)ivai-iant. 



As a corollary we may add: 



There are 00 ' invariant hyperbolic paraboloids through any point in 

 space. 



(Through a point 00 ' rays may be drawn which with the rays of 

 invariant points are parallel to the same plane). 



Again: 



/// the involution of the second kind in space there are 00 ^ invariant 

 hyperbolic paraboloids 



It must be remarked that the above propositions are true for a 



