938 
of the (n—4) points V. By this the rays of a pencil (J/) are arranged 
into a correspondence with characteristic numbers 3(62—11)(n—4) 
and 6(2—4)(7?+4n—7). Observing that the 6rn(n—3) fourpoint 
tangents, which meet in J/, represent (n—4) coincidences each, we 
find for the coincidences U — V the number 
(n—A) [38(6n—11) + 6(n?+-4n—7) —6n(n—8) ] =15(n—4)(4n—5). This 
is therefore the number of curves c* with a jivepoint tangent t,. 
Let us now consider the correspondence between two points 
V,,V., which lie on the same tangent wu. Using the correspondence 
arising between the rays MV, MV, we find in a similar way for 
the number of coincidences V, = V, 12(n?+4n—7)(n—4)(n—5)—6n(n—9) 
(n—4)(n—5) = 6(n—4)(n—5)(n? +11n—14). With this the number of 
curves of N has been found, which are in possession of a tangent 
tys, consequently of a point of undulation, the tangent of which 
touches the curve moreover. 
7. The involution of the second rank, which .V determines on a 
straight line /, has 2(n—2)(n—3) groups, each of which possesses 
two double elements; / is therefore bitangent for as many curves 
of the net. If / rotates round a point P, the points of contact 
R,R’ will describe a curve, which passes (2 —3)(n-+-4) times through 
P; for P as base-point of a net lies on (m—3)(n-+4) curves, which 
are each touched in / by one of their bitangents. From this follows 
that the locus of the pairs R,R’, which we shall indicate by (f)p 
is a curve of order (n—3)(5n—4). 
If we consider the correspondence (AR) on the rays of the 
pencil (P), and, in connection with this, the correspondence between 
the rays MR, MR’, we arrive at the number of coincidences A = R’ 
and we find once more that the fourpoint tangents envelop a curve 
of class 6n(n—8). 
Let us now determine the order of the locus of the groups of 
(n—4) points S, which / has in common with the 2(n—2)(n—3) 
curves cr, for which / is bitangent. The pencil determined by P 
contains 2(n—8)(n—4)(n+1) curves which are cut*) in P by one 
of their bitangents. This number indicates at the same time the 
number of branches of the curve (S)p passing through P; for its 
order we find therefore 2(n—8)(n—4)(n+1)+2(n—2)(n—8)(n—4), or 
2(n—3)(n—4)(2n—1). 
If we associate each point PR to each of the points S belonging 
to the same c”, a correspondence is determined in the pencil of rays 
1) T. p. 102. 
