132 



for it contains one ray of each of the plane pencils (?>). All tangents 

 of the conic o" along wliicli H is intersected by o, belong therefore 

 to [r]. On (f there lies one point i?,* of 6, and one point B* of 

 6,. These two points are also singular, for. the tangent to ö' at 5,* 

 is the image of the line element of N{1,1) that has its null point 

 in fi, ; but this line element is represented by am/ ray of the plane 

 pencil (r) ronnd i?,*. 



The nidi point of the straight line b is represented by tiie plane 

 pencil {0,ui); hence also is a 6'm(/M/r/;-pomi! of the congruence (4,2). 



6. The enveloping cone with vertex F is the image of a system 

 of oo' line elements of which the points P lie on the conic ji\ which 

 is the central projection of the conic p' in the polar plane of F. 

 The straight lines / pass through the projection Q of F. Any line 

 / is tiie projection of a conic through and contains therefore two 

 points P, corresponding to the two points R of (/ in 01. The cone 

 round F has accordingly a system (1,2) for image. The conic Ji' 

 passes through B^ and B„ the |)oiiit Q is to be counted double, 

 being the class curve of /. 



If F describes the straight line /, the corresponding tangent cones 

 form a congruence (2,2) with directriv f. The curves of contact p' 

 pass through the intersections S*, S* of H with the [)olar line of 

 /", and rest on h^ and 6,. Hence the curves /i' form a pencil with 

 the base points B^, B,, S^, S„ which Sive singular null poiiits. Through 

 a point P there passes one line /; for the corresponding point R 

 carries one tangent r that rests on / and has the straight line 

 l^PQ for projection. 



A straight line / defines a point Q of the projection q of f, hence 

 a point F, and through this there pass two tangents r to the conic 

 in 01. The congruence in question (2,2) is therefore represented by 

 a null system iV"(l,2). 



The line ƒ cuts the tangent plane to ^^,6, in a point F*, the 

 projection »S of which lies on h and is a singular null poiiii because 

 the tangent OS represents all line elements round S. 



The intersections F^* and F,* of ƒ and H are singular for the 

 congruence (2,2); their projections F^ and F, on « are therefore 

 singular null points. 



In this way the seven singular null points which ^(1,2) must 

 have '), are indicated. 



1) Gf. e.g. my paper on plane linear null systems. These Proceedings Vol. XV 

 p. 1165. 



