42 



surface with {Fi<j) consists of q and 8 straight lines r,. The eightli 

 of those straight lines belongs to a degenerate q\ which touches ff. 

 improperly. 



The plane <i has in common with *'\ besides two times the curve 

 of contact f/', another curve '/'". which has a sextuple point in /S'„. 

 Outside »S'„ the curves '/" and ^ '" have moreover 6x12 — 5 X 6^ 42 

 points in common; from this it ensues (hat eadi plane is osculated 

 by 21 curves (>\ 



The curve ip" along which the plane »|' is touched by '/'", has 

 in common with 0'', outside the intei'section of ^, moreover 

 6X24 — 5X^6 = 64 points. Ttvo arbitrari/ planes are there/ore 

 touched by 64 curves q\ 



5. Any straight line f, containing three points of a (/', is a sin^ 

 giilar trisecant. For through t passes one R'; the remaining ruled 

 surfaces of the net intersect it therefore in the triplets of an invo- 

 lution so that it is trisecant for oo' curves </. From this it ensues 

 that the singular trisecants form a congruence. As each 9* is inter- 

 sected in each of its points by three trisecants, the congruence \t\ 

 is of order three. 



The fundamental points F are singidar points of \t] ; for each of 

 those points bears x' singular trisecants. The cone Ï. which Xhey 

 form, has in common with the cone §.^, which projects an arbitrary 

 p' out of F, three straight lines t, which are nodal edges of.5i', and 

 further the straight lines to the remaining six points F. From this 

 it ensues that J is a cubic cone. The points F are consequently 

 singular points of the third order for the congruence of rays \f\. 



The trisecants of o' form a ruled surface i^v', on which q^ is 

 a triple curve '). 



The axial ruled surface ~?i formed by the straight lines t. resting 

 on a straight line a has therefore with a o* in common the 24 

 points, in whicli (/ is intersected by the eight trisecants resting on 

 a. Outside these points they have only in common the seven points 

 F, which, however, are threefold on -1. We conclude from this that 

 31 must be a ruled surface of order nine. As n is a triple straight 

 line on it, a jiiane passing through a possesses moreover six straight 

 lines t. The congruence of rays [f] is therefore of class sir. 



In connection with this the plane F^F^F^ contains, besides the 



') The points of support of the trisecants form the pairs of an involulorial 

 correspondence (6). The involiiliou I^, which the planes passing through a straight 

 line I produce on q^, has apparently 24 pairs in common with (6) ; consequently 

 eight trisecants rest on ?. 



