939 
with vertex J/, which correspondence has (m—8) (5n—4) (n—-+) and 
4 (n—3) (n—4) (2n—1) as characteristic numbers. 
Since the ray MP contains 4 (n—2) (n— 3) points R, which are 
each associated to (n—4) points S, so that MP is to be considered 
as 4 (n—2) (n—3) (n—4)-fold coincidence, we find for the number of 
coincidences In (n—-3) (n—-4). By this we again find the class of 
the curve enveloped by the tangents fo3 (cf. $ 3). 
A new result is arrived at from the correspondence between two 
points SS, belonging to the same pair R, Rk’. The symmetrical 
correspondence between the rays J/S,, MS, has as characteristic 
number 2 (2n—1) (n—3) (n— 4) (n—5). Any of the groups of (n-—4) 
points S lying on MP produces (n-—4)n—5) pairs S,,S,, so that 
MP represents 2 (n— 3) (n—A4) (n—5) coincidences. The remaining 
[4 (2n—1) — 2 (n—2)] (n—3) (n—4) (n—5) coincidences are, taken 
three by three, points of contact of triple tangents f222. Through 
an arbitrary point P pass consequently 2n (n-—3) (n—4) (n—5) triple 
tangents. 
8. Let a again be an arbitrary straight line; each of its points 
is, as base-point of a pencil belonging to MN, point of contact R of 
nd) (n— 3) bitangents d'). We determine the order of the locus 
of the second point of contact A’. The latter has in common with 
a the pairs of points R, Rk’, in which a is touched by c”, and also 
the ae of undulation (R’== fF), lying on a, consequently 
4 (n—2) (n—3) + 3 (6n—11) or (4n?—2n—9Y) in all. This number is 
apparently A order of the curve (A), in question. 
In order to determine the locus of the points W, which each 
bitangent d of the system in question has moreover in common 
with the c”, twice touched by it, we associate to each of those 
curves c”, the bitangent d, for which the point of contact 7 lies on a. 
To the pencil, which a point P sets apart from NV, a curve of 
order (n—8) (2n?-+-d5n—6) is associated, which contains the points 
of contact of the bitangents to the curves of that pencil’). By this 
the number of straight lines d becomes known, of which a point 
of contact lies on a; the system [c*| has therefore as index (n—8) 
(2n? + 5n —6). The index of the system |d | is (n—3) (5n—4); for this 
is (§ 7) the ae of intersections of @ with the curve (&)p. The 
systems |c"| and |{d] rendered projective, produce a locus of order 
(n—3) (2n*? + 5n—6) + n (n—8) (5n—4). To it belongs the straight 
1) T. p. 102. 
3) Bitangential curve; cf. T. p. 107. 
