( 87 ) 



Evidently b is nodal righl line of 2? 4 ; for, b is the image of the 

 nodal curve ,i' :i lying on 2'*. 



Through a point S of b pass two curves p* ; their two points of 

 intersection ,S" and S" with ö are the points which b has still in 

 common with the surface A" determined by c, S and the points 0. 



As the pairs of points S and S' form a (2,2) correspondence, four 

 curves q* can be brought through four points, which touch a right 

 line and intersect an other right line. 



The section of 2£ 4 with the plane 0^0, has nodes in (\, 0„ O s 

 and in the intersection with b ; so it consists of two conies. One of 

 these conies contains also the intersection of c; it is completed to a 

 degenerated q 3 by the right line out of < resting upon it and 

 upon b. The second conic contains the intersection of the transversal 

 drawn out of 4 to b and c and forms with this right line a q\ 



On the surface JS 14 lie therefore eight conies, nine simple right lines 

 and a nodal line. 



§ 5. The number of curves ^> 3 through Ok {k = 1, 2, 3, 4) resting 

 on the right lines a, b, e, d is evidently as large as the number of 

 transversals of four cubic curves « 3 ,/:J 3 ,y 3 ,d 3 brought through Ok- The 

 scroll (a, ft /), having a s , |i 3 and a right line / as directrices, is of 

 order 14, / being fivefold and each plane through / containing nine 

 right lines. If /„ passes through O l a plane through / contains but 

 four right lines, so that the order of the scroll {a, ft /„) amounts but 

 to 9. From this ensues that («, ft I) possesses four twofold points Oh 



With y a the scroll («, ft /) has 22 points in common outside Ok ; 

 so («, ft y) is of order 22. 



On the scroll («, ft l ) we find that 1 is fivefold, because a right 

 line through (J l cuts four generatrices; on the other hand 0, , O t 

 and A are threefold points, for a right line through O t cutting the 

 fivefold right line / , meets but one generatrix more. With y 3 the 

 scroll («, ft /„) has still 9 X 3 ~ 5 — 3 X 3 = 13 points in common 

 besides the multiple points. In connection with the above follows 

 from this that 1 is a ninefold point on (a, ft y). 



Of the points of intersection of (a, ft y) with ó 3 36 lie in Ok', 

 consequently a\ /i 3 , y 3 , d 3 have thirty common transversals. 



Therefore we can bring through four points thirty cubic curves 

 resting on four given right lines. 



§ 6. Let us now consider the surface if'" formed by the curves 

 i>* resting on a, b and c. Through a point A of a and the points O 



