1068 
passing through M represent each (n—6) (v-—7) coincidences. From 
the remaining ones we find, that I possesses (n—5) (n—6) (n-—7) 
dn? + 102n —192) curves with a tangent tips. 
12. Any point is, in general, node of one er belonging to T. 
We consider the system of the ct having their node D on a straight 
line a. The straight line connecting D with the arbitrary point P, 
intersects cr moreover in (n—2) points /7. The nodal curves of which 
a point / lies in P belong to the net with base-point P. Now the 
locus J of the nodes of the net (JAacosi’s curve) is a curve of order 
3 (n—1), with node in P. The locus (/) has therefore a 3 (n-—1)-fold 
point in P; so it is of order (4m —5). In each intersection of (/) 
with a, ac” has a node D, of which one of the tangents d passes 
through P. Consequently the locus (D)p of the nodes of which one 
of the tangents passes through P is a curve of order (42—5) having 
a node in P. Hence a straight line passing through P contains 
moreover (4n—7) points D; any straight line is therefore tangent in 
the node for (4n—T7) nodal curves. 
On a straight line / the tangents d of the nodal curves of which 
the node lies on a, determine a symmetrical correspondence (ZL, L’): 
its characteristic number is apparently (42—5). The intersection of 
a and / represents two coincidences, for the ¢”, which has a node 
there, determines two points Z each coinciding with the correspond- 
ing point LZ’. The remaining coincidences are produced by coinciding 
tangents d,d’. So the locus (C) of the cusps (cusp-locus) of Tis a 
curve of order 4(2n—3). 
18. The curves (D)p and (D)q see § 12, have the (4n—7) 
points D in common, for which PQ is one of the tangents. The 
remaining (42—5)? — (4n—7) = 16n? — 44n + 32 intersections are 
nodes of curves ce", of which the lines d and d’ pass through P 
and Q. 
We now consider the system of the nodal c”, of which a tangent 
d passes through 2. The pairs of tangents d,d’ determine on a 
straight line / a correspondence (L, 1’). Any ray d is tangent for 
(4n—7) curves; to its intersection L correspond therefore (4/—7) 
points ZL’. Through L’ pass (167°—44n 4-32) tangents d’; as many 
points £ have been associated to L’. The coincidences of (L, L’) 
form two groups; the first contains the (42—5) points D situated 
on /, for which d passes through P. The remaining ones arise in 
consequence of d’ coinciding with d; the tangents in the cusps of 
the complex envelop therefore a curve of class (16n? —44n+30). 
