{ fiG3 ) 



eonnnon with the given Ij, ; so on d lie as many points of intersec- 

 tion H oi pairs 1,1' of the involution /^,. Each ray tiirongh a point /f 

 belongs to the complex r, likewise each ray in the connecting 

 plane h of the right lines I, I' ; i. o. w. the complex has (n — 2) (p — 1) 

 principal points H and {n — 2) {p — 1) principal planes h. 



2. On an arbitrary plane a a rational curve c" with [ri — l)-fold 

 point D is determined by 9". The rays of the complex lying in a 

 envelop a curve («) of class {n — 1) {p — 1), the curve of involution 

 (director curve) of I'p in which the points of C' are arranged by 

 the given Ip. ') 



So the complex is of order {n — 1) {p — 1). 



The line of intersection of n with a principal plane h being a 

 ray of F, the curve of (he complex («) touches. all principal planes. 



If a is made to pass through a right line / of 9", then («) splits 

 up into the pencils having for vertices the traces L' of the (p — 1) 

 rays conjugate to / and into a curve of order (n — 2) {p — 1), the 

 curve of the involution of T'p on the curve c"~' which n has in 

 common with (>" besides. So a tangent plane of q" is a singular 

 plane of r. 



The singular surface consists of a, scroll and the principal planes. 



When a tangent plane contains one -of the principal points it passes 

 in general through the directrix d, therefore through all princi[)al 

 points. Then (o) splits up into {n — 2) (p — 1) pencils {H) and (p — i) 

 pencils (L'). 



()f the n — 1 generatrices / through a point H, two, lo and /'«, 

 form a pair of Ip. If we bring « through one of the remaining 

 right lines 4 (^'=1 to n — 3), then («) consists of [p — 1) pencils with 

 vertices L'k, the pencil [H) and a curve of class {71— 2) (p — 1)— 1. 



In an arbitrary plane through H the curve of the complex (o) 

 consists of the pencil (//) and a curve of order {n — 1) (p — 1) — 1. 



3. The rays of the complex through an arbitrary point A 

 envelop a cone {A) of order {n — 1) (p — 1) passing through the 

 principal points. 



If A lies on q" cone (A) consists of (p — 1) concentric pencils and 

 a cone of order (n — 2) {j}- 1). 



If we assume A in a principal plane /( then only one pencil 

 separates itself from the cone of the complex. 



1) I'/, lias {n — 1) (p — 1) paii-s in coinniou ".villi the involution In which an 

 arbitrary pencil determines on C". 



