53 



^ 3. Assume au arbitrary plane -t aiul in it a point P. If Q is 

 the point at infinity of a straight line of the plane pencil (P,jr), P(3 

 can only be the axis of an 0" touching y" in its points of inter- 

 section with the polar line q of Q relative to y\ but at the same 

 time the polar plane of /^relative to the same W, must pass through q. 



The 0^'s through /I,- touching y' at its points of intersection with 

 q, form a pencil ; if Q moves along the stiaight line at infinity r 

 of jr, q revolves round the pole R of r relative to y'. We get in 

 this way oo* 0''s cutting V^ in a system of oo* conies /t" toucliing 

 y' at its intersections with a ray of the plane pencil I'ound R. 



Now I represent the space of the conies of F^ on a five dimen- 

 sional point-space R^ by considering the coefficients of the equation 

 of a k^ as the homogeneous coordinates of a point in R^; to a 

 conic k^ and to a linear system of oo* conies {k')k of Fqq there 

 correspond a point and a linear space /?/,• of R^ and inversely. 



The double straight lines of a (/c*)^ of Fqq envelop a conic; two 

 of those double lines pass through R, hence the image of all double 

 lines through R has 2 points in common with an arbitrary R^; it 

 is a conic k^*. To y* there coi-responds a point P and to the pencils 

 touching 7* in its points of inteisection with rays of the plane pencil 

 round R, there correspond ^the generatrices of the cone K that has 

 R as vertex and /jj' as directrix. 



All the quadratic surfaces through Ai relative to which F and 

 one of the straight lines q are harmonically conjugated, form a 

 linear system of go' individuals, an (0^)j ; this cuts V^ in a (^'')8 to 

 which there corresponds an R^ in R^. Considering the quadratic 

 surfaces through Ai relative to which P and R and P and T^^q 

 are conjugated, it appears that the R„'s corresponding to all the 

 straight lines </, pass through an R^ and lie in an R^. These R^'s 

 cut the space /^3 in which K lies, in a plane pencil the rays of 

 which by means of the straight lines q are protectively associated 

 to the generatrices of K. It happens three times that the associated 

 elements coincide, hence there exist three O^'s through Ai that have 

 a straight line q as chord of contact and the polar plane of F 

 relative to such an 0" passes through q. To a plane pencil (P, :^) 

 there belong thei-efore three rays of For: 



the complex F of the axes of rotation of the quadratic surfaces 

 of revolution throwjh 4 given points is of the order 3, the complex 

 cones are of the order three, the complex curves of the class three. 



§ 4. Algebraically the order of r may be found by determining 



e.g. the complex cone of an arbitrary point. Willi a view to this 1 



4* 



