157 
(10n?— 32n4-26). To the straight lines d’, which pass through P, 
belong the tangents of the ò7, which has its node in P. Each of 
the remaining (10n?—32n+24) straight lines d’ evidently coincides 
with a ray d, and therefore contains a null-point D, for which the 
two null-rays have coincided. If such a straight line is called a 
double null-ray, it ensues from the above that the double null-rays 
envelop a curve of class 2(n—2) (5n—6) *). 
4. The null-rays d, which have a null-ray D on the straight line 
p, envelop a curve (p) of class (4n—5), which has p as (4n—7)- 
fold tangent. It, therefore, intersects p in (An—5) (4n—6) — (4n—7) 
(4n—6) points, which bear each two coinciding null-rays. 
The locus of the points C, which bear a double null-ray, is, 
therefore, of order 4(2n—3). 
The curve (C) is evidently the locus of the cusps of the complex. 
As the order of (C) may also be determined in another way, it 
appears at the same time that the curve (p) has no other multiple 
tangents. 
5. The case n=2 deserves a separate treatment. In the first 
place each line d has now only one null-point; this is the node of 
the conic, which is indicated by three points of d. 
The locus (C) is now of the fourth order, and consists of four 
straight lines cz. For, if the two straight lines of a nodal c’ coincide, 
Cx is a double line. The complex contains, therefore, four double 
Lines, and they are at the same time singular null-rays. 
The vertices S,, of the complete quadrilateral formed by them are 
the singular points of the null-system. 
The curves (p), and (q),, cf. $ 4, have, besides the null-rays of 
the point pg, seven tangents in common, which have each a null- 
point on p and a null-point on qg, and are consequently singular 
null-rays. To them belong the four straight lines cy. Each of the 
remaining three singular null-rays s must belong to oo’ nodal 
conics. S,, bears as singular point, oo * pairs of lines, which form 
an involution of rays; so S,, S,, belongs to two, and then to oo *, 
pairs of lines and consequently must be singular. The diagonals of 
the quadrilateral, which is formed by the four straight lines c, are 
consequently the three singular null-rays required. 
') In other words, the cuspidal tangents of the cuspidal curves of a complex 
envelop a curve of class 2(n—2)(5”—6). In my paper on the characteristic 
numbers of a complex (These Proceedings, Vol. XVII, page 1055, § 13) the 
influence of the critical points in the determination of the class has been overlooked. 
ji lg 
