738 



determines on t' the point T of the trisecant with which it forms 

 a degenerate q'^ (§ 4). 



The curve (i\, which has / as a bisecant belongs to two points 

 of /, and is consequently a twofold carve of A\ 



The locus of the 9" intersected by / is ihevQÏOYQ Si surf ace 0/ order 

 nine with a twofold curve q*i, a triple curve q^ and two straight 

 lines / and t. 



9. A plane through / intersects A' in a curve X^ ; the latter has 

 the two intersections of qU and six points B in common with /; in 

 each point R, X is touched by a q\ 



The points in which a plane is touched by curves q^ lie therefore 

 on a curve y" ; it is the curve of coincidences of the quadruple 

 involution Q\ in which the plane ). is intersected b) the congruences 



The five intersections St of q^ with A ai-e apparently singular 

 points of Q^ ; to aS^ are namely conjugated 00' triplets of points, 

 lying on the cubic curve ö\-, with double point Sk, in which the 

 monoid 4*' (with vertex Sk) is intersected by A. In Sk X is therefore 

 touched by two (>'^ ; the curve of coincidences y" has consequently 

 nodes in each of the five points St, and in Sk the same tangents 



as (7^■^ 



Any point D of the conic (P through Sk is the intersection of a 

 trisecant t, consequently determines a quadruple, of which the 

 remaining three points are produced by the intersection of the curve 

 y^ coupled with t. On the section / of (p we have therefore a cubic 

 involution i^\ of which the groups are completed into quadruples 

 of Q" by the points D. It is evident that Q\ as long as A remains 

 an arbiti'ary plane, cannot possess any other coUinear triplets. 



In each of the points of intersection 1\, T^ of ƒ with t' (^ 4) a 

 t is cu-t by a y% consequently these points are coincidences of the 

 Q\ The remaining coincidences, lying on ƒ, belong to the involution 

 i^«, from this appears again that the order of the curve of coinci- 

 dences is six. 



As the singular point S^ lies on 6' and therefore may be considered 

 as a point D, the curve o^^ is intersected by ƒ in a triplet of the 

 cubic involution /l^ of which the groups are completed into quad- 

 ruples of Q" by *Si. As Yi" cannot possess a second collinear 

 triplet, it is not a central involution ; so it can be determined in 

 00' ways by a pencil of conies of which the base points are /S\, 

 an arbitrary point of o^\ and moreover two points of the line ƒ. 



