942 
moreover in common with f23, we determine the order of the figure 
produced by the projective systems [c”| and [f23|. The former has 
as index 3 (n—8) (n—A4) (n’?+6n—4) i.e. the number of c” with a f23 
appearing in a pencil’) of N. The index of. [t3] is (§ 3) On (n—3) 
(n—4). The figure produced contains the curve (A) twice, the curve 
(/) three times. For the order of S we find therefore 
3(n—3)n—4)(n? + 6n—A4)-+ In? (n—3)(n—4)—6(n—4)(2n? + 5n —12)— 
—-9(n—4)(n? + 6n—13)=3(n —4)(n—5)(4n? + 7n—15). 
By means of this result we can determine the number of twice 
osculating lines ¢33. For this purpose we consider the correspondence 
(MR, MS). Its characteristic numbers are 
3 (2n?+5n— 12) (n—A4) (n—5) and 3 (4n?+7n—15) n—A4) (n—5S). 
Each of the 9n (n—®8) (n—4) f23 belonging to the pencil (M/) is 
(n—5)-fold coincidence, hence the number of coincidences RS is 
(n—4)(n—5) [(6n?-+-15n— 36)4-(12n?+-21n—45)—9n (n—3) | = (n—4) 
(n—5) (9n?-+63n—81). But then the number of twice osculating 
tangents t33 amounts to 2(n—4) (n—5) (n° +7n—9). 
By means of the correspondence between the points / and S of 
the tangents f23 we can find back the number of tangents to, found 
already in § 6. Analogously we obtain by means of the correspond- 
ence between two points S of the same f3 again the number of 
tangents f293 found in $ 11. 
18. If the net has a base-point B, the curves er, having an 
inflection in B are cut by their stationary tangents ¢ in groups of 
(n—3) points 7, lying on a curve (7’)"+% with sextuple point B 
(§ 5). This curve is of class (n+3)(n+2) —30; through L pass 
(n?-+5n—36) of its tangents. In the point of contact A of such a 
tangent the latter is touched by a c”, which it osculates in B; 
consequently B is a (n—4)(n-+9)-fold point on the curve (Das. 
The curves cr, which touch in B at a ray d, form a pencil, con- 
sequently determine on d an involution of order (n—2). As it possesses 
2‘n—3) coineidences there are 2(n—8) c’, which have d as bitangent, 
of which B is one of the points of contact. The second point of 
contact, R, coincides with B if d becomes fourpoint tangent, con- 
sequently B point of undulation. This occurs six times; hence the 
locus (Rs of the points R is a curve of order 2n, with sextuple 
point B. 
Every straight line d cuts the cv, which it touches in B and in 
R, moreover in (n—4) points S. In order to determine the locus 
“nT. p. 106. 
