30H 



null-points A'" of t. Tlie involiilory relation wliicli arises in conse- 

 quence of this in the plane pencil T has as characteristic nnmber 

 \{m — 2){ni' -\- ^m -\- 1)- {ni -\- ]).y,](?/<— 3) ; any i-ay t passing throngh 

 T represents now (in — 2) (m — 3) coincidences. The rennaining coin- 

 cidences to the number of 2(/;i— 3) [{iii — 2)(//i^ + 4;/i -\- 1) —{in -\- 1)^^ 

 — (/,j,2 _|_ ^^i — 2 — 6-2) ('" — 2) {in — 3) form pairs of double null-points. 



There are conseqnenlly \[in — 2) [m — 3)(?/7'' -f- 7//i -[- 4) — ^{ni — 3) 

 {m. -J- 4),s', rdi/s lohick each beur tino douhle null-points. 



A iinll-system ^(1,/y?) with {in' -{- in -\- \) simple singnlar points 

 has therefore 3(/«— 2j (2/// -f- Ij niill-ra,^s with a tri|)le nnll-point 

 and \{ni — 2) (//i — ^){ni- -\- 1 m -\- 4k) null-rays with two double null- 

 points. 



With this the results of § 4 are confirmed. 



For the null-system .H(l,2r — 2) mentioned above .v, 1= r' ; the 

 number of triple null-points amounts therefore to 3(7?''' — 22r-|-12). 

 For r = 3 we find from this 27. For each pencil (c*) each base- 

 point is point of inflexion on three curves c^ ; the number 27 conse- 

 quently arises from the fact that the 9 base-points serve each on 

 three null-rays as triple null-point. As this observation holds good 

 for each pencil (c" j the number of points ^V'^) outside the base- 

 points will be e(jual to 3(6r'^ — 22/' -f- 12). In such a point a c"" has 

 four coinciding points in common with its tangent, In general a 

 pencil (C) has therefore 6(/' — ^3) (3/- — 2) curves tiiat have &. point of 

 undulation ^). 



7. If the curves .4^. =r (^ 1) have an 9*-fold point in 0,, 

 aSo ^ 0^ is an r-fohl wdl-point on each of its rays. Outside the 

 singular null-point S^ there are then moreover (??/" -\- in + 1) — /'' 

 simple singular null-points <S. 



The null-cni've of *S'o has as equation A^ .v^ — A,x^ = 0; hence it 

 has in S^ an (r -\- l)-fol(l point. 



The null-curve {P)"'+^ has in .So an ?'-fold point, consequently 

 sends through P {711* -\- m— 2) — (r" — r) tangents t, of which the 

 points of contact lie on the curve of coincidence y. The latter has 

 nodes in the points S\ so of its intersections with (P)'«+i there lie 

 in S, 3m {m. + 1) — {ni' -\-in — 2 — r' + r) — ■2(///' + m + 1 -?•') = 

 = (3r— l)r points. 



From this it ensues that y has in aS, a (3r — l)-fold point. 



In order to determine the order of the com[)lementary curve, we 

 consider two pencils of null-curves (Ci"'+^) and (^3'"+^), and associate 



i) Another deduction of this nmnber I gave in "h'aisceaux de courbes planes". 

 (Archives Teyler, sér. II, t. Xi, p. 99). 



