57 



tlireefold raj of l] hence the third phiiie pencil in rt hay likewise 

 Di as vertex; the complex rays in .t foi-ni accordingly a plane 

 pencil ronnd ilA and a plane pencil ronnd Di which is to be counted 

 double. 



§ i4. Consider an arbiti-ary plane rr Ihrongh one of the points 

 Di ; in tiiis there lies a plaiu pencil of complex rajs I'onnd /),■, 

 while the rest of the rajs envelop a parabola. Out of each point P 

 of the / of .T there can be drawn besides / one more tangent 

 to the parabola; P is the point of contact if this straight line coin- 

 cides with /^. The plane of the pencil of complex I'ajs through P 

 passes in this case through M and through /),, hence through ??i,, 

 but then P coincides with Di or: 



in a plane through one of the points Di (// to a straight line nii) 

 the complex rags consist of a plane pencil ronnd this point and of 

 the tangents of a parabola with axis jj nii. 



^ J5. In a plane through M theie lies a plane pencil of rajs 

 round this point and as the l^ of this plane p is a double raj of 

 r there lie 2 more plane pencils with centres P on p. The points 

 P and the straight lines p aie conjugated in a null sjstem [2,1]. 

 Bj conjugating to each othei- the points P Ijing on the same straight 

 line, an involution of pairs [2] arises. This involution is quadratic, 

 for on an arbitrarj straight line there lies one pair of conjugated 

 points. 



The involution [2] is not a quadratic inversion as the joins of 

 conjugated points do not pass through a lixed point; consequentlj 

 [2] consists of the pairs of points conjugated to each other relative 

 to the conies of a pencil. This involution has 4 double points (the 

 base points of the pencil), in this case the points Z),, and 3 cardinal 

 points, the diagonal points of the complete quadrangle of the base 

 points, in our case the points //,-, hence: 



the complex r consists of pairs of plane pencils of parallel rays 

 Iging in planes through M. The vertices of the two plane pencils 

 hjing in the same plane, are conjugated points of a quadratic invo- 

 lution in V ^. 



^ 16. If a straight line p of V ,^ revolves round one of its points 

 0, the points associated to p in the mdl sjstem [2,1] describe a 

 curve k* of the 3rd order; this curve passes through 0,{\\yo\i^\\ Hj 

 and touches the straight lines ODi at Z),-. The curves ^'* belonging 

 to all the plane pencils of F^, form a net with seven base points, 

 Hj and Z),. 



