525 



the surface of the tangents of -/. The order of tlie hitter surface, 

 tiiat has y as a double curve (cuspidal curve), is equal to 



We tind this by substituting in the formula ii (7i — 1) — 2/; for n 

 the order c(-\-ii of y and for h the above mentioned number of 

 apparent double points of this curve. As v^ cuts the surface under 

 consideration on the double curve y in [i points, we tind for the 

 number of points of intersection outside y, i. e. the order of the 

 focal surface of the congruence F: 



2|J («—1) — 2r. 



The class of the focal surface of P is equal to Ihe number of 

 planes thi'ough a containing two straight lines of r, hence also of 

 ii, that are infinilel^y neai' to each othei', or equal to the number 

 of planes through p^ touching y outside />,. ^s jj, cuts the curve y 

 in (I points, we find for the class in question : 



2a ((i— 1) — 2r. 



§ 5. In order to find the ordei' of the surface foinied by the ver- 

 tices of the plane pencils containing three generatrices of r, we 

 try to find the number of these plane pencils that have their verti- 

 ces on n. These belong to C and are represented on the trisecants 

 of y that cut v^ outside this curve. 



The order of the surface J of the trisecants of y is found by 

 substituting in the formula: 



{n-2)\h-^n{n-l)l , 

 given by CaYi.ey, for ii the order n -\- [i of y and for h the number 

 of apparent double points of this curve found in ^ 3. We find in 

 this case: 



(« 4- iJ-2) i r + k «(«-!) + è ii (i^- J ) - i (" + i3) (« + i3— 1) ! 

 or, after a simple reduction: 



•(« + /?— 2) 7- -f i « («-1) («-2) + I ^ (/Ï— 1) (/?- 2). 

 In order to find the number of generatrices of d that cut v^, we 

 remark that these are the common straight lines of J and the special 

 linear complex that has v^ as axis. Now the axis of a special linear 

 complex C may be considered as a double line of C. This follows 

 in the first place fiom the representation of C on a hypercone K 

 that has been described in § 2 and through which the axis of 6' is 

 transformed into the vertex of K, but also from the well known 

 property that 7i — 2 generatrices of a scroll of the order n cut a 

 straight line of this scroll. As further ;>, has ^ points in common 



