299 



Any net-curve contains a point P, for wliich it serves as curve 

 {Py. For the Jacohian y% at the same time curve of coincidences 

 of the /,, has nodes in Sk and intersects a c'' of tlie net consequently 

 moreover in 10 points R, which must be coincidences of the 7,. 

 Let R^ be one of those points; the tangent in R^ at c'' has two 

 luoi-e points in common witii that curve ; one of them forms with 

 R^ a triplet of /. I^et P be the second of those |)oints. The {]*y 

 belonging to P has now in common with c* the 13 points »S', the 

 point /-* and the triplet of the /, determined by R^ ; but the two 

 curves are identical then and the tangents at c'^ meet in the 10 

 points R in P. 



From this it ensues at the same time that the lines t containing 

 the coinciclences of /, envelop a curve t of the tenth class. 



10. In Sjc six tangents of aj/ meet ; each of the tangents in Sk 

 replaces two straight lines t, so that t has a node in Sk- 



If {Py has a node D, PI) replaces two straight lines t and P 

 is a point of t. 



If {Py has two nodes Z>, and D,, Pis node of t and J'D„ PD, 

 are the tangents in P. 



Analogously t has a cusp in P, if (/-*)'' is a cuspidal c*. 



(Jonsequently r has besides the 13 nodes Sk, moreover 225 nodes 

 and 72 cusps. ^) 



Hence we find further that t 'is vi curve of order 27 and ot\/eni(s 15. 



It must ' correspond in genus to the curve of coincidence y'; in 

 fact the latter is also of genus 15, because it has J 3 nodes. 



As o'^k contains six coincidences besides Sk, the co)nplementary 

 curve X has a sextuple point in S/c. On each {Py lie 10 points of 

 X, viz. on the straight lines t, which meet in P. So {Py and x have 

 10 -J- J3 X Ö points in common, x is consequently a curve of order 22. 



The curves y' and x^'^ can only touch outside the points aS'; and in 

 each of those points of contact the curves of a pencil (c"*) have an 

 osculation. Fiom 9 >, 22 — 13 \ 2 X 6 = 42 it appears therefore 

 that 7, has 21 groups of which, the three points have coincided. 



11. Let us now consider the case that the cjirves indicated in 

 ^ 9 by a\ = 0, b\ = 0, c%. = have a node in *SV The net [c*] 

 may now be represented by 



') A net [c] without multiple base points has "/j {n—l)(n—^)(Sn^—Sn — l\) 

 binodal and 12 (n — !)(« — 2) cuspidal curves. (Gf. e.g. my paper. in volume VII, 

 p. 631 633). 



