41 



The curves (;>' lying on 2" are represented \)y a pencil of rational 

 curves y" passing through the seven points F'k and thrice through 

 the intersection Q of q. To that pencil belong seven surfaces each 

 consisting of a straight line QF'k and a nodal ff' passing through 

 the remaining points F'. Such a figure is the image of a degenerate 

 q\ of which the q* passes througli S; while the straight line r is 

 produced by the intersection of the plane {Fkq)- 



4. The surface A formed by the curves o\ which intersect a 

 straight line /, has q as a se.vtnple straight line ; for in its intersections 

 with a monoid 2' the line / meets six curves q'' passing through 

 the vertex »S' of the monoid. 



The section of A with the plane {F^ q) consists of tiie sextuple 

 straight line q and three straight lines i\; of these, one is intersected 

 by /, the other two are indicated by the two curves q^\ which 

 rest on / (^2). Tlie surface A is therefore of order nine; it has 

 seven triple points Fk, and contains 21 straight lines r. 



The order of A may also be determined as follows. As in § 3 

 I consider two pencils (/?'). If each two ruled surfaces intersecting 

 on / are associated to each other, a correspondence (3,3) arises. The 

 figure produced by it is of order 18 and consists of three times 

 the ruled surface wiiich tiie pencils have in common and the sur- 

 face A; this surface is consequently of order nine. 



A plane P. passing through / intersects J' along a curve A\ The 

 curve i>\ which has I as a chord (hence is nodal curve of .1') passes 

 through two of the intersections of / and A*; in each of the remain- 

 ing six intersections ^ is touclied by a o\ The locus of the points 

 in which a plane q is touched by curves <>'' is therefore a curve 

 of order sir, q', with quinttiple point S^^^ {q,q). 



With an arbitrary surface A" this curve has, outside *S„, 6 X 9 — 

 5 X 6 = 24 points in common. The curves touching a plane f/ form 

 therefore a surface of order 24, *". 



A monoid ^' has with (f'\ outside -S,, moreover 6x6 — 5X4 

 = 16 points in common; on 7" lie therefore the points of contact 

 of 16 curves 9* passing through the vertex of il*, in other words 

 'f has q as sixteen/old straight line. 



An arbitrary <>* therefore intersects 'P''^ 64 times on q ; as the 

 remaining 56 intersections are united in tiie points F, *" has seven 

 octuple points F. 



The hyperboloid R'^ has, outside *S'„, seven points in common with 

 qp' ; in those 'points <i is louclied by as many rational curves q'. 

 The corresponding straight lines r^ lie on 0". The section of this 



