130 



plane pencil {0,u)) is accordingly the image of the null sjslem ^ (0,1), 

 in which iV lies on b. 



Let (/j be a straight line of H cutting b^ and a, so that its 

 projection i^, passes through Ö,. As anj point of </, ma^' be considered 

 as a point of contact R, P is an arbitrary point of _</, and </, is the 

 image of all line elements lying on (/i. The straigiit lines of the 

 scrolls (<7,) and \ij^) are therefore sim/tilar tangents. 



3. Let the symbol (A, -t) indicate a system of line elements f^,/) 

 in which the points P lie on a curve of the order jt and the 

 straight lines / envelop a curve ot the class i.. 



The image of a plane pencil (1,0) is apjjarently a plane pencil 

 of tangents. If P lies in A, the plane pencil (/) coincides with the 

 plane pencil [A, I). The plane pencils {B^, I) and (i^,, /) are repre- 

 sented by congruences (1,1) (cf. § 2). 



The image of a system (0,1) consists of the tangents of a conic 

 i' lying in the projecting plane of the fixed straight line /. 



A system (1,1) consists of the line elements of wliich P lies on 

 a straight line c and / passes through a point D. If P moves on 

 the straight line c, R describes a conic y' (through (>) and ^ envelops 

 tlie tangent cone which has the pole of the plane y of y' as vertex. 

 The |)lane ö^ 01 revolves round d^ OD and describes a pencil 

 which is projective with the system of the tangent planes q (index 

 2). The image lines r describe accordingly a ciibic scroll of which 

 d is the double directrix and y^ a director curve. 



The intersection of this scroll (/)' and the plane y consists evidently 

 of the conic y' and the tangent o which rests on c and is the 

 image of the line element {B, b) belonging to (1,1). The points of 

 intersection of y' with c lie on the straight lines rt, and a,\ the 

 line elements to which they belong, are represented by the tangents 

 of (/•)' which, apart from o, rest on c. To (?•)' there belong two 

 straight lines of H\ they cut each other on d, and are the images 

 of the line elements for which / passes through B^ or B^. 



4. Let a system (A,jr) be given. The curve [P] which is of the 

 order .t, is projected out of by a cone of the same order, which 

 cuts H along a curve [R) of the order 2n (with a double point in 

 0). The polar plane of the point F, chosen at random, contains 

 accordingly 2/i points R; hence the tangent planes q envelop a 

 surface of the class 2ji. To each plane q there corresponds one 

 plane {01); inversely to one plane 01 (containing .t points P) there 



