526 



-.1 ■. • .1 /?((?— 1)(,^— 2) .... . , , rr^ 



with y, it IS apparently a ; told generatrix of d. llie 



number of generatrices of zJ cutting v^, iw therefore found by 

 diuiinishing the order-number found above, by : 



|/J((i-l)(.i-2). 



Hence tliere are 



(« + ^-2) r + I « («— 1) («-2) 



straight lines of d whicli cut ?;,. 



, «(«f— 1)(((— 2) 

 111 the first place tiie straight line /*, must be oounled ; 



times, for as this line has ^ points in common with y it is an 



fold generatrix of J. Further the number found above 



6 



has to be diminished by the number of trisecants of y tliat cut v^ 



on y. This is the case in each of the /J points (hat y has in common 



with t;,. We find the number of trisecants of y passing through 



sucli a point, by the aid of the property that through a jioint of a 



twisted curve of flie order n witli h apparent double points, there 



pass h — w -|- 2 straight lines that contain two more points of the 



curve, if we lake into account that in our case for each of the 



said /? points Wj counts times among the trisecants of y 



z 



passing through them, as t\ contains ;J — 1 more points of y outside 



the point under consideration. Consequently 



/:? i /• + 1 « («-J) + \ ,:? 0:f- 1) _ « — ^ + 2 — i ((J-1) (;i— 2) j 

 or 



i?!r+|«(«-l)(«-2)! 



trisecants of y that cut <;, on y, must be taken apart. 



If we subtract these two numbers of straight lines from the 

 aforesaid number of straight lines of A that cut u,, we find that 



|(«_2)i6y-(«-lM3(?-l)! 



trisecants of y intersect i\ outside this curve. 



According to the beginning of this § we arrive at the following 

 theorem : 



The locus of the vertices of the plane pencils that have three 

 straight lines in common loith a congruence |<f, (i| of the rank r, is 

 a surface of the order: 



I («_2) |6r - («-1) (3/J-«)!. 



