1065 
curves in (4n—7) (n—3) points G. As base-point of a net belonging 
to I P lies on (6n?—15n-+3) tangents d of nodal curves passing 
through P; so the locus (G) has in P a (6n? 15n-+-3)-fold point 
and is therefore of order (6n?—15n-+3) + (4n—7) (n—3) = 10n? — 
— 34n + 24. 
The correspondence (MD, MG) has as characteristic numbers 
(4n— 5)(n—3) and (10n’—34n+24); the ray MP represents (42 —7) 
(n—8) coincidences. As the remaining ones arise from coincidences 
D=G, it ensues that the inflectional tangents of the flecnodes envelop 
a curve of class (10n?—82n+18), 
16. Let the complex be given by the equation 
aA + pB+ yC + dD=0. 
If the derivative of A with regard to ep is indicated by Aj it 
ensues from the equations 
aA, + BB, + yCe+ dD, =0 (k=1,2,3) 
that an arbitrary point is node of one c”, unless 
ede 
| 
| A, B, 
be Bern de 
be satisfied. 
The exceptional points in question K (critical points) are conse- 
quently common points of the four curves of Jacost belonging to 
the=weis a = 07.6 =O = 0, ¢ = 0: 
To the intersections of |A;,5;,C, —=0 with BeCy Dy =0 belong 
the points, for which we have 
lie! ZA ot Smee 2 on j 
| =e) 
| C, C, oF | 
and they are not situated on the two other curves /. The last 
mentioned relation is apparently satisfied by 2?(m—1)?— (n—1)?= 
3(n—1)’ points; consequently the number of critical points amounts 
to 3° (n—1)*? — 3 (n—1)’ or 6 (n—1)’. 
17. If Fr has a base-point B this is as base-point of any net of 
FP, node of the curves J, consequently represents four points K. 
The number of critical points of a complex with b  base-points 
amounts therefore to 6 (n—1)*—4b. 
Any point K is node of o' curves forming a pencil, hence cusp 
of two curves; the cuspidal tangents are the double rays of the 
