732 



project this curve out of tlie pole F Ijing on i(, consequently form 

 a cone of order n{n-\-i). From tliis follows that thea.vesof\)'"\fonn 

 a complex of rays of order ?z(?i+l). 



In iiny plane passing through a cardinal point H lies a y", which 

 passes through H. The oo^ y" passing through H form therefore a 

 special congruence [/"], which has i/ as fundcimental point; the 

 singular curve o of this congruence has therefore a triple point in 

 7/ (§ 5) ; it is the ö's which has Hk as pole. 



Each point H is triple point of a singular curve o, tahich passes 

 through the remaining cardinal points. 



This curve is base curve of a net of surfaces 2E, which have all 

 a node in H. 



The planes of the nodal curves y",; envelop a surface of class 

 (/z — 1)^(71 4-3), for this is the number of tangent planes of — ^''+'5 

 which pass through a straight lijie r (§ 4). 



The curves y".; form apparently a congruence of which the order 

 and class are [n — 'iy{n-];-Z). 



= 0. 



9. We now assume a tetrahedron of coordinates and consider 



(he net of surfaces ^ belonging to the straight lines of the plane 



x^ = 0. This net may then be represented by 



d» ... 



= 0. 

 4 



The cardinal points are therefore found from 



«"a b'\ c"x d'\ 



^\ '"^'a '^'3 '''A 



From this ensues readily that the curves of the complex may be 

 represented by the relations: 



If we consider here a, ^, y as given, but d as variable, then there 

 arises by elimination of ó the above mentioned equation of the surface 

 ^ belonging to the straight line .i\=0 , ax^-]^ti(.\-\-y.z\ = 0. 



For the curves passing through a point F is 



V 



aa» + 3b'' + re" + rM" =r and ^ ay^ = 0. 

 // .'/ ,'/ // 



By elimination of a,i3,Y,(f out of these equations and :i'aa"=0, 

 2: ad\ = 0, we find for the surface 2»'+^ belonging to Y, the 

 equation 



The a.vis of Y is indicated by 



1 1 y^ a" .i\ 11=0. 



a" X, 



a" 



X 



= 0. 



a" 



V 



