1061 
We now consider again the figure produced by the projective 
systems [c"] and [422]. The former has as index $(n—4)(n—-5)(3n?-+- 
+5n—14) '), the latter, see § 5, 6(n—4)(n—5)(n—1). As the figure 
produced consists of 4(2-+3)(n—4)(n—5) times the line a ®, twice 
the curve (7), and the locus of the points Q, which each c” has 
moreover in common with its 4292, we find for the order of (Q), 
ty (n—d) (Bn? + 5n—14) + 6(n —4) (n—5) (n—1) n—4 (n-- 4) 
(n—5) (n-+3) — 2 (n—5) (12n?—10n—42) or + (n—5)(n—6) (210? — 
Lin --72). : 
The curve (Ca is cut by a in 4(n—3)(n—4)(n—5) groups of 
(n—6) points Q, which are each to be counted thrice, and in a 
number of points (7), where a c”is osculated by a tangent ts. 
From 4 (n—5)(n-—6)(21n?—11n—72) — 4 (n—8) (n—A4) (n—5)(n —6) 
ensues again (§ 6), that the points of contact T, of the tangents 
i322 are situated on a curve of order 4 (n—5)(n—6)(138n?+45n—168). 
The correspondence between the points 7, outside a, and the 
corresponding points Q, produces again the order of the curve (2,) 
belonging to the tangents f293 ($ 7). 
The symmetrical correspondence (MQ, MQ) has as characteristic 
number 4(2— 5)(n—6)(n—7)(21n?—11n—72) and in each be passing 
through J/ (n—6)(n—7) coincidences. From (2—5)(n—6)(n—7)| (210? — 
14n—72)—6(n—1)(n —4)| ensues that the locus of the points of 
contact of the quadruple tangents is a curve of order 5 (n——5)(n—6) 
(n—7) (15n?+19n—96). 
9. Let us now consider the figure determined by the projectivity 
between the curves c”, which possess a f2222 and those quadruple 
tangents. The system [c”] has as index (n—1)(n-+-4) (n—4) (n—5) 
(n—6)(n—7) *), the tangents f.222 form ($ 5) a system with index 
$n(n—4)(n—5)(n—6)(n—7). The figure produced consists of twice 
the locus of the points of contact ($8) and the curve (S) of the 
intersections of the ct with its quadruple tangents. For the order 
of (S) we now find £(n—4)(n—5)(n—6)(n—7)( Tn? +-9n—12)—4 (n—5) 
(n—6) (n—7) (80n? + 38n—192) or 4 (n—5)(n—6)(n—7)(n—8)(7n? + 
in—30). 
The correspondence (7, S) determines in the pencil of rays (J/) 
a correspondence with characteristic numbers }(2—95)(7-—6)(n—7) 
(n--8\(157?+19n—96 and 4(n—5\(n—6)(n—7)(n—8)(7n? + 7n—380). 
As the tangents f022 passing through J/ each represent 4(n—8) 
